| L(s) = 1 | − 3.50·2-s + 3·3-s + 4.29·4-s + 10.2·5-s − 10.5·6-s + 1.41·7-s + 12.9·8-s + 9·9-s − 35.9·10-s + 11·11-s + 12.8·12-s + 13.7·13-s − 4.97·14-s + 30.7·15-s − 79.9·16-s − 65.5·17-s − 31.5·18-s − 118.·19-s + 44.0·20-s + 4.25·21-s − 38.5·22-s − 182.·23-s + 38.9·24-s − 20.0·25-s − 48.2·26-s + 27·27-s + 6.09·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.577·3-s + 0.537·4-s + 0.916·5-s − 0.715·6-s + 0.0765·7-s + 0.573·8-s + 0.333·9-s − 1.13·10-s + 0.301·11-s + 0.310·12-s + 0.293·13-s − 0.0949·14-s + 0.529·15-s − 1.24·16-s − 0.935·17-s − 0.413·18-s − 1.43·19-s + 0.492·20-s + 0.0441·21-s − 0.373·22-s − 1.65·23-s + 0.331·24-s − 0.160·25-s − 0.363·26-s + 0.192·27-s + 0.0411·28-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
| \(L(2)\) |
\(\approx\) |
\(1.391938112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391938112\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.50T + 8T^{2} \) |
| 5 | \( 1 - 10.2T + 125T^{2} \) |
| 7 | \( 1 - 1.41T + 343T^{2} \) |
| 13 | \( 1 - 13.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 65.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 179.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 169.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 36.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 244.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 486.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 208.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 193.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 55.6T + 2.05e5T^{2} \) |
| 67 | \( 1 + 537.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 473.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 996.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 132.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 962.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−9.006141084453965228235434951181, −8.163209453012734656693537288511, −7.59064957436056171276939402389, −6.50958175905185250918656428322, −5.96581345591539370966116321661, −4.57627576448499980329430594252, −3.88099104880391176991205327413, −2.17978375105281710579766210924, −1.98887827515254874817830243110, −0.62744917603441679740321827411,
0.62744917603441679740321827411, 1.98887827515254874817830243110, 2.17978375105281710579766210924, 3.88099104880391176991205327413, 4.57627576448499980329430594252, 5.96581345591539370966116321661, 6.50958175905185250918656428322, 7.59064957436056171276939402389, 8.163209453012734656693537288511, 9.006141084453965228235434951181
