| L(s) = 1 | + 2·3-s − 8·9-s + 36·13-s − 12·19-s + 44·25-s − 22·27-s + 136·31-s − 16·37-s + 72·39-s + 160·43-s + 14·49-s − 24·57-s + 156·61-s + 24·67-s − 32·73-s + 88·75-s + 128·79-s + 7·81-s + 272·93-s − 8·97-s − 352·103-s − 256·109-s − 32·111-s − 288·117-s + 428·121-s + 127-s + 320·129-s + ⋯ |
| L(s) = 1 | + 2/3·3-s − 8/9·9-s + 2.76·13-s − 0.631·19-s + 1.75·25-s − 0.814·27-s + 4.38·31-s − 0.432·37-s + 1.84·39-s + 3.72·43-s + 2/7·49-s − 0.421·57-s + 2.55·61-s + 0.358·67-s − 0.438·73-s + 1.17·75-s + 1.62·79-s + 7/81·81-s + 2.92·93-s − 0.0824·97-s − 3.41·103-s − 2.34·109-s − 0.288·111-s − 2.46·117-s + 3.53·121-s + 0.00787·127-s + 2.48·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(14.67841057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.67841057\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 4 p T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 44 T^{2} + 1034 T^{4} - 44 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 428 T^{2} + 74630 T^{4} - 428 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 18 T + 412 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 988 T^{2} + 407046 T^{4} - 988 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 556 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1444 T^{2} + 980166 T^{4} - 1444 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 2972 T^{2} + 3611558 T^{4} - 2972 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 68 T + 3050 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 1382 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 1292 T^{2} + 2832038 T^{4} - 1292 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 80 T + 5046 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 6148 T^{2} + 19144326 T^{4} - 6148 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} - 13350138 T^{4} + 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 3676 T^{2} + 15964266 T^{4} - 3676 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 78 T + 8620 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 8566 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 10588 T^{2} + 78813510 T^{4} - 10588 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 5990 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 64 T + 4434 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 13948 T^{2} + 141899946 T^{4} - 13948 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 11468 T^{2} + 120945830 T^{4} - 11468 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 18374 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.58202274164577097686986392647, −6.42964763465501524379673729507, −6.25293887218820143154995600103, −6.08114760229647795429654170361, −5.78064601267665451605016038151, −5.63434940554510741316319170455, −5.44731153125577370598158828743, −5.04231425063808481734025521263, −4.99618446851153533571083061154, −4.41091813377127825630965893297, −4.27095739429517601965337149893, −4.20179064838208547084971939518, −4.05568677919573735909092903878, −3.63320970625435457393229937649, −3.29933081880371573659819282622, −3.14532177755826426681166259217, −2.79614731523327093838925229113, −2.63683552739105504748806695233, −2.57628558395088126864059826883, −2.02658835640111546943122020019, −1.77815118065169332903092205983, −1.20756976373146912393641328859, −0.947611530417737044759561518104, −0.70317908654106436638004781304, −0.61063241991143029314758919049,
0.61063241991143029314758919049, 0.70317908654106436638004781304, 0.947611530417737044759561518104, 1.20756976373146912393641328859, 1.77815118065169332903092205983, 2.02658835640111546943122020019, 2.57628558395088126864059826883, 2.63683552739105504748806695233, 2.79614731523327093838925229113, 3.14532177755826426681166259217, 3.29933081880371573659819282622, 3.63320970625435457393229937649, 4.05568677919573735909092903878, 4.20179064838208547084971939518, 4.27095739429517601965337149893, 4.41091813377127825630965893297, 4.99618446851153533571083061154, 5.04231425063808481734025521263, 5.44731153125577370598158828743, 5.63434940554510741316319170455, 5.78064601267665451605016038151, 6.08114760229647795429654170361, 6.25293887218820143154995600103, 6.42964763465501524379673729507, 6.58202274164577097686986392647
