| L(s) = 1 | − 2-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s − 4·13-s + 14-s − 16-s − 3·17-s − 5·19-s − 2·20-s + 3·23-s − 25-s + 4·26-s + 28-s − 29-s − 9·31-s − 5·32-s + 3·34-s − 2·35-s − 2·37-s + 5·38-s + 6·40-s − 5·41-s − 8·43-s − 3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s − 1.10·13-s + 0.267·14-s − 1/4·16-s − 0.727·17-s − 1.14·19-s − 0.447·20-s + 0.625·23-s − 1/5·25-s + 0.784·26-s + 0.188·28-s − 0.185·29-s − 1.61·31-s − 0.883·32-s + 0.514·34-s − 0.338·35-s − 0.328·37-s + 0.811·38-s + 0.948·40-s − 0.780·41-s − 1.21·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.24916615430618, −12.91463280084416, −12.37680423521939, −11.78039894441897, −11.16755004557339, −10.68567107950482, −10.24455034589534, −9.925846049548499, −9.310793905875559, −9.197367806497776, −8.537263245618396, −8.227790338496359, −7.480766931802484, −6.978011263747079, −6.745513981372832, −5.925296084073016, −5.516221686914024, −4.900129704504418, −4.579995870682128, −3.844342152227679, −3.340863992708544, −2.472392604843230, −1.960407005563943, −1.624417117545082, −0.5578303912675606, 0,
0.5578303912675606, 1.624417117545082, 1.960407005563943, 2.472392604843230, 3.340863992708544, 3.844342152227679, 4.579995870682128, 4.900129704504418, 5.516221686914024, 5.925296084073016, 6.745513981372832, 6.978011263747079, 7.480766931802484, 8.227790338496359, 8.537263245618396, 9.197367806497776, 9.310793905875559, 9.925846049548499, 10.24455034589534, 10.68567107950482, 11.16755004557339, 11.78039894441897, 12.37680423521939, 12.91463280084416, 13.24916615430618
