| L(s) = 1 | + 1.08·2-s − 3·3-s − 6.82·4-s + 5·5-s − 3.24·6-s + 24.7·7-s − 16.0·8-s + 9·9-s + 5.41·10-s − 28.5·11-s + 20.4·12-s + 10.7·13-s + 26.8·14-s − 15·15-s + 37.2·16-s + 133.·17-s + 9.73·18-s + 88.3·19-s − 34.1·20-s − 74.3·21-s − 30.9·22-s − 30.3·23-s + 48.1·24-s + 25·25-s + 11.6·26-s − 27·27-s − 169.·28-s + ⋯ |
| L(s) = 1 | + 0.382·2-s − 0.577·3-s − 0.853·4-s + 0.447·5-s − 0.220·6-s + 1.33·7-s − 0.709·8-s + 0.333·9-s + 0.171·10-s − 0.783·11-s + 0.492·12-s + 0.229·13-s + 0.511·14-s − 0.258·15-s + 0.582·16-s + 1.90·17-s + 0.127·18-s + 1.06·19-s − 0.381·20-s − 0.772·21-s − 0.299·22-s − 0.274·23-s + 0.409·24-s + 0.200·25-s + 0.0879·26-s − 0.192·27-s − 1.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.429342377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.429342377\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 163 | \( 1 - 163T \) |
| good | 2 | \( 1 - 1.08T + 8T^{2} \) |
| 7 | \( 1 - 24.7T + 343T^{2} \) |
| 11 | \( 1 + 28.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 133.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 161.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 106.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 294.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 216.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 269.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 416.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 733.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 23.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 552.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 765.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 571.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 452.T + 9.12e5T^{2} \) |
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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
−8.280785101341156002861429054149, −8.068994312668644218624102428658, −7.08733106304958908958620997533, −5.77347789012205993014771023733, −5.41287193149622476986976937339, −4.89593654140981777610714465727, −3.92117757811624025908119740741, −2.94134459514353097683343941968, −1.55369194982788515020295804172, −0.73310426445531602991784311596,
0.73310426445531602991784311596, 1.55369194982788515020295804172, 2.94134459514353097683343941968, 3.92117757811624025908119740741, 4.89593654140981777610714465727, 5.41287193149622476986976937339, 5.77347789012205993014771023733, 7.08733106304958908958620997533, 8.068994312668644218624102428658, 8.280785101341156002861429054149
