| L(s) = 1 | + 2-s + 1.74·3-s + 4-s + 0.258·5-s + 1.74·6-s − 2.13·7-s + 8-s + 0.0291·9-s + 0.258·10-s + 1.46·11-s + 1.74·12-s − 6.38·13-s − 2.13·14-s + 0.449·15-s + 16-s − 0.785·17-s + 0.0291·18-s − 1.66·19-s + 0.258·20-s − 3.70·21-s + 1.46·22-s + 6.44·23-s + 1.74·24-s − 4.93·25-s − 6.38·26-s − 5.17·27-s − 2.13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.00·3-s + 0.5·4-s + 0.115·5-s + 0.710·6-s − 0.805·7-s + 0.353·8-s + 0.00973·9-s + 0.0815·10-s + 0.440·11-s + 0.502·12-s − 1.77·13-s − 0.569·14-s + 0.115·15-s + 0.250·16-s − 0.190·17-s + 0.00688·18-s − 0.383·19-s + 0.0576·20-s − 0.809·21-s + 0.311·22-s + 1.34·23-s + 0.355·24-s − 0.986·25-s − 1.25·26-s − 0.995·27-s − 0.402·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 1.74T + 3T^{2} \) |
| 5 | \( 1 - 0.258T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 6.38T + 13T^{2} \) |
| 17 | \( 1 + 0.785T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 37 | \( 1 + 1.48T + 37T^{2} \) |
| 41 | \( 1 + 6.43T + 41T^{2} \) |
| 43 | \( 1 - 0.972T + 43T^{2} \) |
| 47 | \( 1 + 9.94T + 47T^{2} \) |
| 53 | \( 1 - 1.91T + 53T^{2} \) |
| 59 | \( 1 - 2.02T + 59T^{2} \) |
| 61 | \( 1 - 2.52T + 61T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 + 0.963T + 71T^{2} \) |
| 73 | \( 1 - 6.89T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 2.31T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
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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
−7.59202315338983012377464559275, −7.02150054287567320267729331882, −6.38671778876693231576515941983, −5.46254361557349134672571277531, −4.79633949649795416139510275624, −3.86869382407511102348613908618, −3.19302050780537154411860051382, −2.56146921884984116557539171729, −1.79975259686339162157454531197, 0,
1.79975259686339162157454531197, 2.56146921884984116557539171729, 3.19302050780537154411860051382, 3.86869382407511102348613908618, 4.79633949649795416139510275624, 5.46254361557349134672571277531, 6.38671778876693231576515941983, 7.02150054287567320267729331882, 7.59202315338983012377464559275
