| L(s) = 1 | + 2-s + 4-s − 3.93·5-s − 1.01·7-s + 8-s − 3.93·10-s − 0.888·11-s + 3.28·13-s − 1.01·14-s + 16-s + 4.43·17-s − 0.450·19-s − 3.93·20-s − 0.888·22-s − 1.53·23-s + 10.4·25-s + 3.28·26-s − 1.01·28-s − 4.45·29-s + 0.873·31-s + 32-s + 4.43·34-s + 3.98·35-s + 1.43·37-s − 0.450·38-s − 3.93·40-s − 0.965·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.76·5-s − 0.382·7-s + 0.353·8-s − 1.24·10-s − 0.268·11-s + 0.912·13-s − 0.270·14-s + 0.250·16-s + 1.07·17-s − 0.103·19-s − 0.880·20-s − 0.189·22-s − 0.319·23-s + 2.09·25-s + 0.644·26-s − 0.191·28-s − 0.826·29-s + 0.156·31-s + 0.176·32-s + 0.760·34-s + 0.673·35-s + 0.236·37-s − 0.0730·38-s − 0.622·40-s − 0.150·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 229 | \( 1 + T \) |
| good | 5 | \( 1 + 3.93T + 5T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 + 0.888T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 19 | \( 1 + 0.450T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 + 4.45T + 29T^{2} \) |
| 31 | \( 1 - 0.873T + 31T^{2} \) |
| 37 | \( 1 - 1.43T + 37T^{2} \) |
| 41 | \( 1 + 0.965T + 41T^{2} \) |
| 43 | \( 1 - 0.0256T + 43T^{2} \) |
| 47 | \( 1 + 5.37T + 47T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 + 6.20T + 61T^{2} \) |
| 67 | \( 1 + 9.43T + 67T^{2} \) |
| 71 | \( 1 - 3.44T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 1.20T + 79T^{2} \) |
| 83 | \( 1 - 4.61T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 3.53T + 97T^{2} \) |
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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
−7.925767797324255649634630698536, −7.40891337747822194391904835977, −6.57097373213777047789369241708, −5.81784230116038475693110685178, −4.89828327352799851379214666293, −4.11806205390268248834419582927, −3.49296485334040426175314514661, −2.95130579237992180174849632669, −1.38621184095445609801691195308, 0,
1.38621184095445609801691195308, 2.95130579237992180174849632669, 3.49296485334040426175314514661, 4.11806205390268248834419582927, 4.89828327352799851379214666293, 5.81784230116038475693110685178, 6.57097373213777047789369241708, 7.40891337747822194391904835977, 7.925767797324255649634630698536
