| L(s) = 1 | − 0.375·2-s − 1.85·4-s − 2.49·5-s + 7-s + 1.44·8-s + 0.937·10-s + 0.0439·11-s + 1.23·13-s − 0.375·14-s + 3.17·16-s + 4.66·17-s − 4.55·19-s + 4.64·20-s − 0.0165·22-s + 5.31·23-s + 1.23·25-s − 0.464·26-s − 1.85·28-s + 5.23·29-s + 0.715·31-s − 4.08·32-s − 1.75·34-s − 2.49·35-s + 7.90·37-s + 1.71·38-s − 3.61·40-s + 4.97·41-s + ⋯ |
| L(s) = 1 | − 0.265·2-s − 0.929·4-s − 1.11·5-s + 0.377·7-s + 0.512·8-s + 0.296·10-s + 0.0132·11-s + 0.343·13-s − 0.100·14-s + 0.793·16-s + 1.13·17-s − 1.04·19-s + 1.03·20-s − 0.00352·22-s + 1.10·23-s + 0.246·25-s − 0.0910·26-s − 0.351·28-s + 0.972·29-s + 0.128·31-s − 0.722·32-s − 0.300·34-s − 0.422·35-s + 1.29·37-s + 0.277·38-s − 0.571·40-s + 0.776·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.044874412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.044874412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.375T + 2T^{2} \) |
| 5 | \( 1 + 2.49T + 5T^{2} \) |
| 11 | \( 1 - 0.0439T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 4.66T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 - 5.31T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 - 0.715T + 31T^{2} \) |
| 37 | \( 1 - 7.90T + 37T^{2} \) |
| 41 | \( 1 - 4.97T + 41T^{2} \) |
| 43 | \( 1 - 0.341T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 4.41T + 53T^{2} \) |
| 59 | \( 1 + 9.78T + 59T^{2} \) |
| 61 | \( 1 + 9.42T + 61T^{2} \) |
| 67 | \( 1 - 0.00811T + 67T^{2} \) |
| 71 | \( 1 - 4.91T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 1.44T + 79T^{2} \) |
| 83 | \( 1 - 7.62T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 8.97T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.943896249985666134069360278077, −7.47021372755660764272389679624, −6.49256729561760140429733346089, −5.69735795254282918864311380570, −4.75555200675748272651887489255, −4.40587130108324231817481704279, −3.60925332306397224581786097317, −2.88106038226220591508230866474, −1.43598822903198532335978528267, −0.58210249404207636364466936864,
0.58210249404207636364466936864, 1.43598822903198532335978528267, 2.88106038226220591508230866474, 3.60925332306397224581786097317, 4.40587130108324231817481704279, 4.75555200675748272651887489255, 5.69735795254282918864311380570, 6.49256729561760140429733346089, 7.47021372755660764272389679624, 7.943896249985666134069360278077
