| L(s) = 1 | − 0.494·2-s − 1.75·4-s + 5-s − 4.28·7-s + 1.85·8-s − 0.494·10-s − 4.48·11-s − 13-s + 2.11·14-s + 2.59·16-s − 1.57·17-s + 1.86·19-s − 1.75·20-s + 2.21·22-s + 5.25·23-s + 25-s + 0.494·26-s + 7.52·28-s + 0.750·29-s + 10.2·31-s − 4.99·32-s + 0.776·34-s − 4.28·35-s + 2.78·37-s − 0.923·38-s + 1.85·40-s + 6.36·41-s + ⋯ |
| L(s) = 1 | − 0.349·2-s − 0.877·4-s + 0.447·5-s − 1.61·7-s + 0.656·8-s − 0.156·10-s − 1.35·11-s − 0.277·13-s + 0.565·14-s + 0.648·16-s − 0.381·17-s + 0.428·19-s − 0.392·20-s + 0.472·22-s + 1.09·23-s + 0.200·25-s + 0.0969·26-s + 1.42·28-s + 0.139·29-s + 1.83·31-s − 0.882·32-s + 0.133·34-s − 0.724·35-s + 0.457·37-s − 0.149·38-s + 0.293·40-s + 0.993·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.494T + 2T^{2} \) |
| 7 | \( 1 + 4.28T + 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 17 | \( 1 + 1.57T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 - 5.25T + 23T^{2} \) |
| 29 | \( 1 - 0.750T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 - 8.27T + 47T^{2} \) |
| 53 | \( 1 + 0.0752T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 9.37T + 73T^{2} \) |
| 79 | \( 1 + 6.09T + 79T^{2} \) |
| 83 | \( 1 + 0.612T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.77422851541333187883223475935, −7.31263265742717876279510911568, −6.29059148982833266916014371246, −5.76135485824667989525809939651, −4.87825901260352586735954631375, −4.23141216008912418457220035304, −2.99847867665541398806009586615, −2.68156309043291675796941682119, −1.01268224970995548806560796035, 0,
1.01268224970995548806560796035, 2.68156309043291675796941682119, 2.99847867665541398806009586615, 4.23141216008912418457220035304, 4.87825901260352586735954631375, 5.76135485824667989525809939651, 6.29059148982833266916014371246, 7.31263265742717876279510911568, 7.77422851541333187883223475935
