| L(s) = 1 | − 1.44·2-s + 0.493·3-s + 0.0775·4-s − 3.65·5-s − 0.711·6-s + 2.24·7-s + 2.77·8-s − 2.75·9-s + 5.26·10-s − 4.20·11-s + 0.0382·12-s + 4.42·13-s − 3.24·14-s − 1.80·15-s − 4.14·16-s + 6.53·17-s + 3.97·18-s − 19-s − 0.282·20-s + 1.10·21-s + 6.06·22-s + 7.00·23-s + 1.36·24-s + 8.33·25-s − 6.37·26-s − 2.84·27-s + 0.174·28-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.284·3-s + 0.0387·4-s − 1.63·5-s − 0.290·6-s + 0.849·7-s + 0.979·8-s − 0.918·9-s + 1.66·10-s − 1.26·11-s + 0.0110·12-s + 1.22·13-s − 0.866·14-s − 0.465·15-s − 1.03·16-s + 1.58·17-s + 0.936·18-s − 0.229·19-s − 0.0632·20-s + 0.242·21-s + 1.29·22-s + 1.46·23-s + 0.279·24-s + 1.66·25-s − 1.24·26-s − 0.546·27-s + 0.0329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6596658779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6596658779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 + 1.44T + 2T^{2} \) |
| 3 | \( 1 - 0.493T + 3T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 4.42T + 13T^{2} \) |
| 17 | \( 1 - 6.53T + 17T^{2} \) |
| 23 | \( 1 - 7.00T + 23T^{2} \) |
| 29 | \( 1 + 0.884T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 43 | \( 1 - 3.71T + 43T^{2} \) |
| 47 | \( 1 - 8.75T + 47T^{2} \) |
| 53 | \( 1 + 2.75T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 6.88T + 79T^{2} \) |
| 83 | \( 1 - 7.85T + 83T^{2} \) |
| 89 | \( 1 + 4.28T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.062077390229326984708783548899, −7.64044714544052129724045251259, −7.31905558654605743082340114887, −5.87992662983836504188393433681, −5.12622882435247395361320065631, −4.42314578201690270362358311436, −3.50990762625453375972571674669, −2.89315869616112501308593017090, −1.46455291011137869615677733354, −0.52955696939849382817615453299,
0.52955696939849382817615453299, 1.46455291011137869615677733354, 2.89315869616112501308593017090, 3.50990762625453375972571674669, 4.42314578201690270362358311436, 5.12622882435247395361320065631, 5.87992662983836504188393433681, 7.31905558654605743082340114887, 7.64044714544052129724045251259, 8.062077390229326984708783548899
