L(s) = 1 | + 1.54·3-s − 0.356·5-s − 7-s − 0.600·9-s − 11-s − 13-s − 0.552·15-s − 6.04·17-s + 8.48·19-s − 1.54·21-s + 7.98·23-s − 4.87·25-s − 5.57·27-s + 5.31·29-s − 9.77·31-s − 1.54·33-s + 0.356·35-s + 10.5·37-s − 1.54·39-s + 8.72·41-s − 7.46·43-s + 0.214·45-s + 1.45·47-s + 49-s − 9.37·51-s − 1.71·53-s + 0.356·55-s + ⋯ |
L(s) = 1 | + 0.894·3-s − 0.159·5-s − 0.377·7-s − 0.200·9-s − 0.301·11-s − 0.277·13-s − 0.142·15-s − 1.46·17-s + 1.94·19-s − 0.338·21-s + 1.66·23-s − 0.974·25-s − 1.07·27-s + 0.987·29-s − 1.75·31-s − 0.269·33-s + 0.0603·35-s + 1.73·37-s − 0.248·39-s + 1.36·41-s − 1.13·43-s + 0.0319·45-s + 0.212·47-s + 0.142·49-s − 1.31·51-s − 0.235·53-s + 0.0481·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 1.54T + 3T^{2} \) |
| 5 | \( 1 + 0.356T + 5T^{2} \) |
| 17 | \( 1 + 6.04T + 17T^{2} \) |
| 19 | \( 1 - 8.48T + 19T^{2} \) |
| 23 | \( 1 - 7.98T + 23T^{2} \) |
| 29 | \( 1 - 5.31T + 29T^{2} \) |
| 31 | \( 1 + 9.77T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 - 1.45T + 47T^{2} \) |
| 53 | \( 1 + 1.71T + 53T^{2} \) |
| 59 | \( 1 + 0.781T + 59T^{2} \) |
| 61 | \( 1 + 5.31T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 + 7.65T + 71T^{2} \) |
| 73 | \( 1 + 2.44T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 - 9.25T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−7.51256788851546266981464638341, −7.05442874029935066663612203746, −6.08926878088758594031440321289, −5.39329961839413794906042713564, −4.59374936830261858522298320030, −3.75789773961424168077913420177, −2.91481707441893768589152126901, −2.57731296082003771467397346684, −1.35214261081046446943092972678, 0,
1.35214261081046446943092972678, 2.57731296082003771467397346684, 2.91481707441893768589152126901, 3.75789773961424168077913420177, 4.59374936830261858522298320030, 5.39329961839413794906042713564, 6.08926878088758594031440321289, 7.05442874029935066663612203746, 7.51256788851546266981464638341