| L(s) = 1 | − 2·2-s − 9.48·3-s + 4·4-s + 18.9·6-s + 8.59·7-s − 8·8-s + 62.9·9-s + 44.0·11-s − 37.9·12-s + 8.11·13-s − 17.1·14-s + 16·16-s − 87.4·17-s − 125.·18-s − 150.·19-s − 81.5·21-s − 88.1·22-s − 23·23-s + 75.8·24-s − 16.2·26-s − 341.·27-s + 34.3·28-s + 85.9·29-s + 209.·31-s − 32·32-s − 417.·33-s + 174.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.82·3-s + 0.5·4-s + 1.29·6-s + 0.464·7-s − 0.353·8-s + 2.33·9-s + 1.20·11-s − 0.912·12-s + 0.173·13-s − 0.328·14-s + 0.250·16-s − 1.24·17-s − 1.64·18-s − 1.81·19-s − 0.847·21-s − 0.854·22-s − 0.208·23-s + 0.645·24-s − 0.122·26-s − 2.43·27-s + 0.232·28-s + 0.550·29-s + 1.21·31-s − 0.176·32-s − 2.20·33-s + 0.882·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 9.48T + 27T^{2} \) |
| 7 | \( 1 - 8.59T + 343T^{2} \) |
| 11 | \( 1 - 44.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.11T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 150.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 85.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 38.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 399.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 127.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 594.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 459.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 582.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 344.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 478.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 726.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 475.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 374.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 152.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 497.T + 9.12e5T^{2} \) |
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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
−9.038400374780202520180759079979, −8.299657843534570558022649738368, −7.00009510102525626048437687522, −6.52144071287263437745785734950, −5.92094340353061622819243376531, −4.69049530202617096648720489480, −4.10792048175304136694517745843, −2.07108592130977408733327682841, −1.05639391955626002714414976428, 0,
1.05639391955626002714414976428, 2.07108592130977408733327682841, 4.10792048175304136694517745843, 4.69049530202617096648720489480, 5.92094340353061622819243376531, 6.52144071287263437745785734950, 7.00009510102525626048437687522, 8.299657843534570558022649738368, 9.038400374780202520180759079979
