| L(s) = 1 | − 8·2-s + 93·3-s + 64·4-s − 744·6-s − 343·7-s − 512·8-s + 6.46e3·9-s − 2.16e3·11-s + 5.95e3·12-s + 1.66e3·13-s + 2.74e3·14-s + 4.09e3·16-s + 3.57e4·17-s − 5.16e4·18-s + 2.02e4·19-s − 3.18e4·21-s + 1.73e4·22-s + 4.21e4·23-s − 4.76e4·24-s − 1.32e4·26-s + 3.97e5·27-s − 2.19e4·28-s − 1.11e5·29-s − 2.69e5·31-s − 3.27e4·32-s − 2.01e5·33-s − 2.86e5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.98·3-s + 1/2·4-s − 1.40·6-s − 0.377·7-s − 0.353·8-s + 2.95·9-s − 0.490·11-s + 0.994·12-s + 0.209·13-s + 0.267·14-s + 1/4·16-s + 1.76·17-s − 2.08·18-s + 0.676·19-s − 0.751·21-s + 0.347·22-s + 0.722·23-s − 0.703·24-s − 0.148·26-s + 3.88·27-s − 0.188·28-s − 0.851·29-s − 1.62·31-s − 0.176·32-s − 0.976·33-s − 1.24·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.872924891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.872924891\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 - 31 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 197 p T + p^{7} T^{2} \) |
| 13 | \( 1 - 1661 T + p^{7} T^{2} \) |
| 17 | \( 1 - 35771 T + p^{7} T^{2} \) |
| 19 | \( 1 - 20222 T + p^{7} T^{2} \) |
| 23 | \( 1 - 42130 T + p^{7} T^{2} \) |
| 29 | \( 1 + 111789 T + p^{7} T^{2} \) |
| 31 | \( 1 + 269504 T + p^{7} T^{2} \) |
| 37 | \( 1 + 532774 T + p^{7} T^{2} \) |
| 41 | \( 1 - 158056 T + p^{7} T^{2} \) |
| 43 | \( 1 - 521874 T + p^{7} T^{2} \) |
| 47 | \( 1 - 939733 T + p^{7} T^{2} \) |
| 53 | \( 1 - 408384 T + p^{7} T^{2} \) |
| 59 | \( 1 + 522172 T + p^{7} T^{2} \) |
| 61 | \( 1 - 350080 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3931176 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1194016 T + p^{7} T^{2} \) |
| 73 | \( 1 + 998350 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2120709 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1746708 T + p^{7} T^{2} \) |
| 89 | \( 1 + 10077740 T + p^{7} T^{2} \) |
| 97 | \( 1 - 6238295 T + p^{7} T^{2} \) |
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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
−9.912607781928938345564656479641, −9.264804343018053835880600098964, −8.548968692271623092341896054123, −7.53609532903041297872526797399, −7.21470041543639929219315915528, −5.45854214692887209322650504832, −3.72402278165236025195009713669, −3.11572443822819559663958822200, −2.03841423362161299395608595339, −0.989096433350369858435886063749,
0.989096433350369858435886063749, 2.03841423362161299395608595339, 3.11572443822819559663958822200, 3.72402278165236025195009713669, 5.45854214692887209322650504832, 7.21470041543639929219315915528, 7.53609532903041297872526797399, 8.548968692271623092341896054123, 9.264804343018053835880600098964, 9.912607781928938345564656479641
