| L(s) = 1 | − 2.09·2-s + 2.37·4-s − 4.22·5-s − 1.68·7-s − 0.783·8-s + 8.83·10-s − 2.06·11-s + 4.77·13-s + 3.52·14-s − 3.11·16-s + 0.783·17-s + 0.328·19-s − 10.0·20-s + 4.32·22-s + 9.05·23-s + 12.8·25-s − 9.98·26-s − 3.99·28-s + 0.158·31-s + 8.07·32-s − 1.63·34-s + 7.11·35-s − 1.11·37-s − 0.688·38-s + 3.30·40-s + 9.30·41-s − 1.00·43-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.18·4-s − 1.88·5-s − 0.636·7-s − 0.276·8-s + 2.79·10-s − 0.622·11-s + 1.32·13-s + 0.941·14-s − 0.777·16-s + 0.189·17-s + 0.0754·19-s − 2.24·20-s + 0.921·22-s + 1.88·23-s + 2.56·25-s − 1.95·26-s − 0.755·28-s + 0.0283·31-s + 1.42·32-s − 0.280·34-s + 1.20·35-s − 0.182·37-s − 0.111·38-s + 0.523·40-s + 1.45·41-s − 0.153·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4815305578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4815305578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 + 1.68T + 7T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 - 0.783T + 17T^{2} \) |
| 19 | \( 1 - 0.328T + 19T^{2} \) |
| 23 | \( 1 - 9.05T + 23T^{2} \) |
| 31 | \( 1 - 0.158T + 31T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 - 9.30T + 41T^{2} \) |
| 43 | \( 1 + 1.00T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 - 4.98T + 67T^{2} \) |
| 71 | \( 1 + 1.82T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 8.47T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 4.67T + 97T^{2} \) |
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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
−8.009527937789017319477025086495, −7.36509134172736299869489217754, −6.96291959543119620363763736922, −6.10678503368465372643847020250, −4.96200740904113207734406063449, −4.18106655826067159313173923705, −3.38099362895615812340330467427, −2.74340347721595922088946699742, −1.20667867883353681301510065083, −0.51335314235516420149401676383,
0.51335314235516420149401676383, 1.20667867883353681301510065083, 2.74340347721595922088946699742, 3.38099362895615812340330467427, 4.18106655826067159313173923705, 4.96200740904113207734406063449, 6.10678503368465372643847020250, 6.96291959543119620363763736922, 7.36509134172736299869489217754, 8.009527937789017319477025086495
