| L(s) = 1 | + 4.15·2-s + 3·3-s + 9.26·4-s + 17.8·5-s + 12.4·6-s − 31.9·7-s + 5.24·8-s + 9·9-s + 74.3·10-s + 0.210·11-s + 27.7·12-s − 3.15·13-s − 132.·14-s + 53.6·15-s − 52.3·16-s + 48.3·17-s + 37.3·18-s + 116.·19-s + 165.·20-s − 95.9·21-s + 0.874·22-s + 23·23-s + 15.7·24-s + 194.·25-s − 13.1·26-s + 27·27-s − 296.·28-s + ⋯ |
| L(s) = 1 | + 1.46·2-s + 0.577·3-s + 1.15·4-s + 1.59·5-s + 0.848·6-s − 1.72·7-s + 0.231·8-s + 0.333·9-s + 2.34·10-s + 0.00577·11-s + 0.668·12-s − 0.0673·13-s − 2.53·14-s + 0.923·15-s − 0.817·16-s + 0.689·17-s + 0.489·18-s + 1.40·19-s + 1.85·20-s − 0.997·21-s + 0.00847·22-s + 0.208·23-s + 0.133·24-s + 1.55·25-s − 0.0989·26-s + 0.192·27-s − 1.99·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.770125671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.770125671\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 4.15T + 8T^{2} \) |
| 5 | \( 1 - 17.8T + 125T^{2} \) |
| 7 | \( 1 + 31.9T + 343T^{2} \) |
| 11 | \( 1 - 0.210T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.15T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 31 | \( 1 - 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 374.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 137.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 519.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 632.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 132.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 154.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 897.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 555.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 623.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 709.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 762.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 367.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 53.6T + 9.12e5T^{2} \) |
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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
−9.249112564388717905293346135124, −7.82749759759419142219744689140, −6.74368527144539697524029085810, −6.22688467971718396809524010045, −5.67996185715651169792710503024, −4.81888564098888690965966850001, −3.69942962693414983688502273954, −2.88701912998077891824880927711, −2.52841470068758670132662051499, −1.02722650447550218109557811478,
1.02722650447550218109557811478, 2.52841470068758670132662051499, 2.88701912998077891824880927711, 3.69942962693414983688502273954, 4.81888564098888690965966850001, 5.67996185715651169792710503024, 6.22688467971718396809524010045, 6.74368527144539697524029085810, 7.82749759759419142219744689140, 9.249112564388717905293346135124
