| L(s) = 1 | + 4·4-s + 2·5-s − 3·9-s − 6·11-s + 4·16-s − 2·19-s + 8·20-s + 25-s + 10·29-s − 16·31-s − 12·36-s − 16·41-s − 24·44-s − 6·45-s − 3·49-s − 12·55-s + 30·59-s − 46·61-s − 4·64-s − 24·71-s − 8·76-s − 8·79-s + 8·80-s + 6·81-s + 42·89-s − 4·95-s + 18·99-s + ⋯ |
| L(s) = 1 | + 2·4-s + 0.894·5-s − 9-s − 1.80·11-s + 16-s − 0.458·19-s + 1.78·20-s + 1/5·25-s + 1.85·29-s − 2.87·31-s − 2·36-s − 2.49·41-s − 3.61·44-s − 0.894·45-s − 3/7·49-s − 1.61·55-s + 3.90·59-s − 5.88·61-s − 1/2·64-s − 2.84·71-s − 0.917·76-s − 0.900·79-s + 0.894·80-s + 2/3·81-s + 4.45·89-s − 0.410·95-s + 1.80·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.127097019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.127097019\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| 11 | \( ( 1 + T )^{6} \) |
| good | 2 | \( 1 - p^{2} T^{2} + 3 p^{2} T^{4} - 7 p^{2} T^{6} + 3 p^{4} T^{8} - p^{6} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 66 T^{2} + 1931 T^{4} - 32288 T^{6} + 1931 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 91 T^{2} + 3614 T^{4} - 80103 T^{6} + 3614 p^{2} T^{8} - 91 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + T + 44 T^{2} + 33 T^{3} + 44 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 119 T^{2} + 6294 T^{4} - 187787 T^{6} + 6294 p^{2} T^{8} - 119 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 5 T + 80 T^{2} - 289 T^{3} + 80 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 8 T + 65 T^{2} + 282 T^{3} + 65 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 210 T^{2} + 18779 T^{4} - 915968 T^{6} + 18779 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 8 T + 141 T^{2} + 666 T^{3} + 141 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 219 T^{2} + 21086 T^{4} - 1162547 T^{6} + 21086 p^{2} T^{8} - 219 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 254 T^{2} + 27975 T^{4} - 1715024 T^{6} + 27975 p^{2} T^{8} - 254 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 139 T^{2} + 10974 T^{4} - 619087 T^{6} + 10974 p^{2} T^{8} - 139 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 15 T + 248 T^{2} - 1877 T^{3} + 248 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 23 T + 314 T^{2} + 2919 T^{3} + 314 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{3} \) |
| 71 | \( ( 1 + 12 T + 191 T^{2} + 1402 T^{3} + 191 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 214 T^{2} + 26303 T^{4} - 2177460 T^{6} + 26303 p^{2} T^{8} - 214 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 4 T + 193 T^{2} + 670 T^{3} + 193 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 199 T^{2} + 21726 T^{4} - 1728427 T^{6} + 21726 p^{2} T^{8} - 199 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 21 T + 4 p T^{2} - 3505 T^{3} + 4 p^{2} T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 463 T^{2} + 98830 T^{4} - 12248383 T^{6} + 98830 p^{2} T^{8} - 463 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.48360041094809188186767334173, −4.95149349207645265797818126183, −4.77015980306479305086737181169, −4.67171883711606721266436460739, −4.60088196102274483627507142139, −4.55878617272930932342601111859, −4.42777835885310003762430264438, −3.87371687907968260713520300786, −3.64716746265621903100955347088, −3.58531341388092293939534308781, −3.43769306012875907659741150430, −3.24999620053839923833270164598, −3.13200741443580217638743461141, −2.93007925624338390692724037891, −2.48739692291699389284034118783, −2.44812840088827538575083986144, −2.44576867813478874526591403358, −2.40298280913271964015211146646, −2.01564658542379749353381285674, −1.76017486186065429908705472011, −1.50205526758356361744039586439, −1.49582256991069986776967106125, −1.18316338219943226111164851371, −0.46147843569848726125009426241, −0.16715875423941808764067629880,
0.16715875423941808764067629880, 0.46147843569848726125009426241, 1.18316338219943226111164851371, 1.49582256991069986776967106125, 1.50205526758356361744039586439, 1.76017486186065429908705472011, 2.01564658542379749353381285674, 2.40298280913271964015211146646, 2.44576867813478874526591403358, 2.44812840088827538575083986144, 2.48739692291699389284034118783, 2.93007925624338390692724037891, 3.13200741443580217638743461141, 3.24999620053839923833270164598, 3.43769306012875907659741150430, 3.58531341388092293939534308781, 3.64716746265621903100955347088, 3.87371687907968260713520300786, 4.42777835885310003762430264438, 4.55878617272930932342601111859, 4.60088196102274483627507142139, 4.67171883711606721266436460739, 4.77015980306479305086737181169, 4.95149349207645265797818126183, 5.48360041094809188186767334173
Plot not available for L-functions of degree greater than 10.
