| L(s) = 1 | + 11·4-s − 148·11-s − 31·16-s − 336·19-s − 664·29-s + 640·31-s + 724·41-s − 1.62e3·44-s − 147·49-s + 360·59-s + 2.44e3·61-s − 865·64-s − 272·71-s − 3.69e3·76-s + 2.06e3·79-s + 48·81-s − 484·89-s + 2.98e3·101-s − 1.90e3·109-s − 7.30e3·116-s + 5.28e3·121-s + 7.04e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 11/8·4-s − 4.05·11-s − 0.484·16-s − 4.05·19-s − 4.25·29-s + 3.70·31-s + 2.75·41-s − 5.57·44-s − 3/7·49-s + 0.794·59-s + 5.12·61-s − 1.68·64-s − 0.454·71-s − 5.57·76-s + 2.94·79-s + 0.0658·81-s − 0.576·89-s + 2.93·101-s − 1.67·109-s − 5.84·116-s + 3.97·121-s + 5.09·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3053084285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3053084285\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| good | 2 | \( 1 - 11 T^{2} + 19 p^{3} T^{4} - 287 p^{2} T^{6} + 19 p^{9} T^{8} - 11 p^{12} T^{10} + p^{18} T^{12} \) |
| 3 | \( 1 - 16 p T^{4} - 32150 T^{6} - 16 p^{7} T^{8} + p^{18} T^{12} \) |
| 11 | \( ( 1 + 74 T + 5570 T^{2} + 204680 T^{3} + 5570 p^{3} T^{4} + 74 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 13 | \( 1 - 328 p T^{2} + 2269272 T^{4} + 11996164138 T^{6} + 2269272 p^{6} T^{8} - 328 p^{13} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 3280 T^{2} - 8890128 T^{4} + 11736667810 T^{6} - 8890128 p^{6} T^{8} - 3280 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 + 168 T + 26197 T^{2} + 2333344 T^{3} + 26197 p^{3} T^{4} + 168 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 61090 T^{2} + 1667208207 T^{4} - 26076219594620 T^{6} + 1667208207 p^{6} T^{8} - 61090 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 + 332 T + 80572 T^{2} + 13628846 T^{3} + 80572 p^{3} T^{4} + 332 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( ( 1 - 320 T + 113341 T^{2} - 19016064 T^{3} + 113341 p^{3} T^{4} - 320 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 276770 T^{2} + 33129403527 T^{4} - 2195897974516860 T^{6} + 33129403527 p^{6} T^{8} - 276770 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 - 362 T + 248299 T^{2} - 51434996 T^{3} + 248299 p^{3} T^{4} - 362 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 297754 T^{2} + 45863741847 T^{4} - 4531415720155052 T^{6} + 45863741847 p^{6} T^{8} - 297754 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 - 322152 T^{2} + 42988826152 T^{4} - 4194937101563966 T^{6} + 42988826152 p^{6} T^{8} - 322152 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 - 199818 T^{2} + 55441412887 T^{4} - 6092351194402124 T^{6} + 55441412887 p^{6} T^{8} - 199818 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 - 180 T + 3577 T^{2} + 128522760 T^{3} + 3577 p^{3} T^{4} - 180 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( ( 1 - 1222 T + 1103759 T^{2} - 593135356 T^{3} + 1103759 p^{3} T^{4} - 1222 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 - 142482 T^{2} + 146353877367 T^{4} - 11082662491564636 T^{6} + 146353877367 p^{6} T^{8} - 142482 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 136 T + 900677 T^{2} + 112926832 T^{3} + 900677 p^{3} T^{4} + 136 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 804458 T^{2} + 433750956447 T^{4} - 209358603495872844 T^{6} + 433750956447 p^{6} T^{8} - 804458 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 - 1034 T + 1094262 T^{2} - 675989052 T^{3} + 1094262 p^{3} T^{4} - 1034 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 1653138 T^{2} + 1219490719447 T^{4} - 685677675218773724 T^{6} + 1219490719447 p^{6} T^{8} - 1653138 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 242 T + 1427227 T^{2} + 347564516 T^{3} + 1427227 p^{3} T^{4} + 242 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 4522112 T^{2} + 9234443526912 T^{4} - 10842927360521653566 T^{6} + 9234443526912 p^{6} T^{8} - 4522112 p^{12} T^{10} + p^{18} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.62778837656947951726706959367, −6.45478885279477600481668469227, −6.22155011683113553807281937385, −5.95154462912950063063656485706, −5.63643084049098432980040717033, −5.47584508609517522446347329847, −5.39887238938654544260999259064, −5.32240696727169126043826565893, −4.82383342464332496869441584053, −4.64597721459979143774824722465, −4.46690025153983776732904224606, −4.15198589252756528746572332693, −4.10282366820149423972329004894, −3.71545346338740674883199952044, −3.50975500326661890101014991270, −2.90382035880224243580856052698, −2.78631820347767917680299181218, −2.54751090868037895702696075158, −2.34990657030382244464280719337, −2.12447001269129357267572721492, −1.98394844495889893801492098413, −1.95362432027385244342434311381, −0.76539158959867827534382511093, −0.65723044471188283264812014818, −0.093442374290872021546514594534,
0.093442374290872021546514594534, 0.65723044471188283264812014818, 0.76539158959867827534382511093, 1.95362432027385244342434311381, 1.98394844495889893801492098413, 2.12447001269129357267572721492, 2.34990657030382244464280719337, 2.54751090868037895702696075158, 2.78631820347767917680299181218, 2.90382035880224243580856052698, 3.50975500326661890101014991270, 3.71545346338740674883199952044, 4.10282366820149423972329004894, 4.15198589252756528746572332693, 4.46690025153983776732904224606, 4.64597721459979143774824722465, 4.82383342464332496869441584053, 5.32240696727169126043826565893, 5.39887238938654544260999259064, 5.47584508609517522446347329847, 5.63643084049098432980040717033, 5.95154462912950063063656485706, 6.22155011683113553807281937385, 6.45478885279477600481668469227, 6.62778837656947951726706959367
Plot not available for L-functions of degree greater than 10.
