| L(s) = 1 | − 4.78·2-s + 6.91·3-s + 14.8·4-s + 9.47·5-s − 33.1·6-s + 13.6·7-s − 32.9·8-s + 20.8·9-s − 45.3·10-s + 11·11-s + 103.·12-s − 65.0·14-s + 65.5·15-s + 38.6·16-s − 124.·17-s − 99.9·18-s + 65.7·19-s + 141.·20-s + 94.1·21-s − 52.6·22-s − 58.7·23-s − 228.·24-s − 35.2·25-s − 42.3·27-s + 202.·28-s + 4.97·29-s − 313.·30-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.33·3-s + 1.86·4-s + 0.847·5-s − 2.25·6-s + 0.734·7-s − 1.45·8-s + 0.773·9-s − 1.43·10-s + 0.301·11-s + 2.47·12-s − 1.24·14-s + 1.12·15-s + 0.604·16-s − 1.77·17-s − 1.30·18-s + 0.793·19-s + 1.57·20-s + 0.978·21-s − 0.510·22-s − 0.532·23-s − 1.94·24-s − 0.282·25-s − 0.301·27-s + 1.36·28-s + 0.0318·29-s − 1.90·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.78T + 8T^{2} \) |
| 3 | \( 1 - 6.91T + 27T^{2} \) |
| 5 | \( 1 - 9.47T + 125T^{2} \) |
| 7 | \( 1 - 13.6T + 343T^{2} \) |
| 17 | \( 1 + 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 4.97T + 2.43e4T^{2} \) |
| 31 | \( 1 + 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 376.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 4.21T + 6.89e4T^{2} \) |
| 43 | \( 1 + 179.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 194.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 336.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 192.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 438.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 164.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 69.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 22.4T + 9.12e5T^{2} \) |
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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
−8.696059357377604595889168533474, −8.019961493802070101583627542725, −7.27407307691885079618005804592, −6.56534753084611119290472097485, −5.40884509474069870781813727617, −4.12024517156775672317901326485, −2.88462778665759713087939989867, −1.90193014598775575471276835820, −1.65790186887438149764031309011, 0,
1.65790186887438149764031309011, 1.90193014598775575471276835820, 2.88462778665759713087939989867, 4.12024517156775672317901326485, 5.40884509474069870781813727617, 6.56534753084611119290472097485, 7.27407307691885079618005804592, 8.019961493802070101583627542725, 8.696059357377604595889168533474
