| L(s) = 1 | − 5.41·2-s + 3·3-s + 21.3·4-s − 13.4·5-s − 16.2·6-s + 11.1·7-s − 72.0·8-s + 9·9-s + 72.8·10-s − 11·11-s + 63.9·12-s + 79.3·13-s − 60.4·14-s − 40.3·15-s + 219.·16-s + 23.6·17-s − 48.7·18-s − 63.5·19-s − 286.·20-s + 33.4·21-s + 59.5·22-s + 208.·23-s − 216.·24-s + 56.3·25-s − 429.·26-s + 27·27-s + 237.·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 0.577·3-s + 2.66·4-s − 1.20·5-s − 1.10·6-s + 0.602·7-s − 3.18·8-s + 0.333·9-s + 2.30·10-s − 0.301·11-s + 1.53·12-s + 1.69·13-s − 1.15·14-s − 0.695·15-s + 3.42·16-s + 0.336·17-s − 0.637·18-s − 0.767·19-s − 3.20·20-s + 0.347·21-s + 0.577·22-s + 1.89·23-s − 1.83·24-s + 0.450·25-s − 3.23·26-s + 0.192·27-s + 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 5.41T + 8T^{2} \) |
| 5 | \( 1 + 13.4T + 125T^{2} \) |
| 7 | \( 1 - 11.1T + 343T^{2} \) |
| 13 | \( 1 - 79.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 63.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 208.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 264.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 99.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 244.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 364.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 462.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 225.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 384.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 715.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 413.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 518.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 656.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 525.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 207.T + 9.12e5T^{2} \) |
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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
−8.606721277981241598391490929267, −7.81962087992314534818372709919, −7.33776082408298891390770426655, −6.55911370954483443623300097326, −5.39043322239765205958690255487, −3.85859609195913582660325977401, −3.19766367299805339431106481444, −1.92801859801381456661624625731, −1.11273846154449489684545355619, 0,
1.11273846154449489684545355619, 1.92801859801381456661624625731, 3.19766367299805339431106481444, 3.85859609195913582660325977401, 5.39043322239765205958690255487, 6.55911370954483443623300097326, 7.33776082408298891390770426655, 7.81962087992314534818372709919, 8.606721277981241598391490929267
