| L(s) = 1 | − 2-s − 0.394·3-s + 4-s + 4.36·5-s + 0.394·6-s + 7-s − 8-s − 2.84·9-s − 4.36·10-s − 1.35·11-s − 0.394·12-s − 3.09·13-s − 14-s − 1.72·15-s + 16-s − 3.52·17-s + 2.84·18-s + 3.98·19-s + 4.36·20-s − 0.394·21-s + 1.35·22-s − 0.903·23-s + 0.394·24-s + 14.0·25-s + 3.09·26-s + 2.30·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.227·3-s + 0.5·4-s + 1.95·5-s + 0.161·6-s + 0.377·7-s − 0.353·8-s − 0.948·9-s − 1.37·10-s − 0.409·11-s − 0.113·12-s − 0.857·13-s − 0.267·14-s − 0.444·15-s + 0.250·16-s − 0.854·17-s + 0.670·18-s + 0.914·19-s + 0.975·20-s − 0.0860·21-s + 0.289·22-s − 0.188·23-s + 0.0805·24-s + 2.80·25-s + 0.606·26-s + 0.443·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.674132265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.674132265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 0.394T + 3T^{2} \) |
| 5 | \( 1 - 4.36T + 5T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 + 0.903T + 23T^{2} \) |
| 29 | \( 1 - 2.54T + 29T^{2} \) |
| 31 | \( 1 + 5.16T + 31T^{2} \) |
| 37 | \( 1 + 9.36T + 37T^{2} \) |
| 41 | \( 1 + 0.259T + 41T^{2} \) |
| 43 | \( 1 - 7.42T + 43T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 7.10T + 59T^{2} \) |
| 61 | \( 1 - 8.93T + 61T^{2} \) |
| 67 | \( 1 + 8.46T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 - 7.04T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 5.97T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−8.286608347879324470266804919747, −7.25172254417942437122914583400, −6.71568265488012696555919467942, −5.88184816936952759440243161699, −5.36811716897018617393317948651, −4.87707952937092304735107588058, −3.29534177640272491911288972986, −2.33223297450216470779477231927, −2.02366827740028985613764781037, −0.74611731791158899513260270205,
0.74611731791158899513260270205, 2.02366827740028985613764781037, 2.33223297450216470779477231927, 3.29534177640272491911288972986, 4.87707952937092304735107588058, 5.36811716897018617393317948651, 5.88184816936952759440243161699, 6.71568265488012696555919467942, 7.25172254417942437122914583400, 8.286608347879324470266804919747
