| L(s) = 1 | + 0.782·2-s − 1.17·3-s − 1.38·4-s − 5-s − 0.919·6-s + 1.16·7-s − 2.65·8-s − 1.62·9-s − 0.782·10-s + 11-s + 1.62·12-s − 5.54·13-s + 0.914·14-s + 1.17·15-s + 0.697·16-s − 3.49·17-s − 1.26·18-s − 2.56·19-s + 1.38·20-s − 1.37·21-s + 0.782·22-s − 6.91·23-s + 3.11·24-s + 25-s − 4.34·26-s + 5.42·27-s − 1.62·28-s + ⋯ |
| L(s) = 1 | + 0.553·2-s − 0.678·3-s − 0.693·4-s − 0.447·5-s − 0.375·6-s + 0.441·7-s − 0.937·8-s − 0.540·9-s − 0.247·10-s + 0.301·11-s + 0.470·12-s − 1.53·13-s + 0.244·14-s + 0.303·15-s + 0.174·16-s − 0.846·17-s − 0.298·18-s − 0.587·19-s + 0.310·20-s − 0.299·21-s + 0.166·22-s − 1.44·23-s + 0.635·24-s + 0.200·25-s − 0.851·26-s + 1.04·27-s − 0.306·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4921516838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4921516838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.782T + 2T^{2} \) |
| 3 | \( 1 + 1.17T + 3T^{2} \) |
| 7 | \( 1 - 1.16T + 7T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 - 6.75T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 + 0.0832T + 41T^{2} \) |
| 43 | \( 1 - 6.14T + 43T^{2} \) |
| 47 | \( 1 - 2.48T + 47T^{2} \) |
| 53 | \( 1 - 0.0722T + 53T^{2} \) |
| 59 | \( 1 + 0.738T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 4.44T + 67T^{2} \) |
| 71 | \( 1 + 0.132T + 71T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 9.19T + 83T^{2} \) |
| 89 | \( 1 - 7.57T + 89T^{2} \) |
| 97 | \( 1 + 8.97T + 97T^{2} \) |
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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
−8.373943776288996383719924828467, −7.78945928090215373106324170369, −6.82236132689805016513142528334, −6.01277552869345592430069738730, −5.41518737238924206412399721077, −4.52938913303437532175322770512, −4.27251066890617284488744128289, −3.09085818265481252919987583550, −2.09690565664649688711514505051, −0.36316894824511124789139339977,
0.36316894824511124789139339977, 2.09690565664649688711514505051, 3.09085818265481252919987583550, 4.27251066890617284488744128289, 4.52938913303437532175322770512, 5.41518737238924206412399721077, 6.01277552869345592430069738730, 6.82236132689805016513142528334, 7.78945928090215373106324170369, 8.373943776288996383719924828467
