| L(s) = 1 | − 2.66·2-s + 5.07·4-s + 5-s + 4.68·7-s − 8.19·8-s − 2.66·10-s − 3.50·11-s − 4.56·13-s − 12.4·14-s + 11.6·16-s + 6.56·17-s − 4.10·19-s + 5.07·20-s + 9.32·22-s + 6.92·23-s + 25-s + 12.1·26-s + 23.7·28-s − 29-s + 8.10·31-s − 14.5·32-s − 17.4·34-s + 4.68·35-s + 4.27·37-s + 10.9·38-s − 8.19·40-s − 5.10·41-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 2.53·4-s + 0.447·5-s + 1.76·7-s − 2.89·8-s − 0.841·10-s − 1.05·11-s − 1.26·13-s − 3.32·14-s + 2.91·16-s + 1.59·17-s − 0.942·19-s + 1.13·20-s + 1.98·22-s + 1.44·23-s + 0.200·25-s + 2.38·26-s + 4.49·28-s − 0.185·29-s + 1.45·31-s − 2.57·32-s − 2.99·34-s + 0.791·35-s + 0.702·37-s + 1.77·38-s − 1.29·40-s − 0.797·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8855067710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8855067710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 7 | \( 1 - 4.68T + 7T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 - 6.56T + 17T^{2} \) |
| 19 | \( 1 + 4.10T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 31 | \( 1 - 8.10T + 31T^{2} \) |
| 37 | \( 1 - 4.27T + 37T^{2} \) |
| 41 | \( 1 + 5.10T + 41T^{2} \) |
| 43 | \( 1 + 8.47T + 43T^{2} \) |
| 47 | \( 1 - 5.41T + 47T^{2} \) |
| 53 | \( 1 - 4.20T + 53T^{2} \) |
| 59 | \( 1 - 5.36T + 59T^{2} \) |
| 61 | \( 1 - 1.27T + 61T^{2} \) |
| 67 | \( 1 - 9.96T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 8.37T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 - 5.46T + 97T^{2} \) |
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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
−9.730955696061759256548690302413, −8.691532990679175642361826726721, −8.104743122307452318525578101850, −7.60314183237129379068350510049, −6.82259209356041568925178562274, −5.52780307981793170979619684669, −4.85244140875330066387011520809, −2.80572240898260933064835461589, −2.02802529977135241119668376759, −0.941004506408326481606102938697,
0.941004506408326481606102938697, 2.02802529977135241119668376759, 2.80572240898260933064835461589, 4.85244140875330066387011520809, 5.52780307981793170979619684669, 6.82259209356041568925178562274, 7.60314183237129379068350510049, 8.104743122307452318525578101850, 8.691532990679175642361826726721, 9.730955696061759256548690302413
