| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 66.4·5-s − 36·6-s + 22.9·7-s + 64·8-s + 81·9-s − 265.·10-s − 294.·11-s − 144·12-s + 180.·13-s + 91.8·14-s + 598.·15-s + 256·16-s + 218.·17-s + 324·18-s − 2.07e3·19-s − 1.06e3·20-s − 206.·21-s − 1.17e3·22-s − 1.86e3·23-s − 576·24-s + 1.29e3·25-s + 720.·26-s − 729·27-s + 367.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.18·5-s − 0.408·6-s + 0.177·7-s + 0.353·8-s + 0.333·9-s − 0.840·10-s − 0.733·11-s − 0.288·12-s + 0.295·13-s + 0.125·14-s + 0.686·15-s + 0.250·16-s + 0.183·17-s + 0.235·18-s − 1.31·19-s − 0.594·20-s − 0.102·21-s − 0.518·22-s − 0.734·23-s − 0.204·24-s + 0.413·25-s + 0.209·26-s − 0.192·27-s + 0.0885·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.739460403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.739460403\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 5 | \( 1 + 66.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 22.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 294.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 180.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 218.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.07e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.86e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.29e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.41e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.72e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 4.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.08e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.34e5T + 8.58e9T^{2} \) |
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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
−10.91994568185974111540631260103, −10.09526867430232319559887434225, −8.451073654705029928416516548437, −7.74952437752571824636457808333, −6.69385899783871538167729195123, −5.68917325821079527112231554061, −4.54643016007894026201367027579, −3.84911495510079868043670026569, −2.40883600999887802840932431599, −0.64870039828973088126366923481,
0.64870039828973088126366923481, 2.40883600999887802840932431599, 3.84911495510079868043670026569, 4.54643016007894026201367027579, 5.68917325821079527112231554061, 6.69385899783871538167729195123, 7.74952437752571824636457808333, 8.451073654705029928416516548437, 10.09526867430232319559887434225, 10.91994568185974111540631260103
