| L(s) = 1 | − 2-s + 1.20·3-s + 4-s − 2.97·5-s − 1.20·6-s − 7-s − 8-s − 1.55·9-s + 2.97·10-s + 4.89·11-s + 1.20·12-s + 0.849·13-s + 14-s − 3.58·15-s + 16-s + 0.911·17-s + 1.55·18-s − 0.375·19-s − 2.97·20-s − 1.20·21-s − 4.89·22-s − 0.153·23-s − 1.20·24-s + 3.86·25-s − 0.849·26-s − 5.47·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.695·3-s + 0.5·4-s − 1.33·5-s − 0.491·6-s − 0.377·7-s − 0.353·8-s − 0.516·9-s + 0.941·10-s + 1.47·11-s + 0.347·12-s + 0.235·13-s + 0.267·14-s − 0.925·15-s + 0.250·16-s + 0.221·17-s + 0.365·18-s − 0.0860·19-s − 0.665·20-s − 0.262·21-s − 1.04·22-s − 0.0320·23-s − 0.245·24-s + 0.773·25-s − 0.166·26-s − 1.05·27-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
| \(L(1)\) |
\(\approx\) |
\(1.127709627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.127709627\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 + 2.97T + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.849T + 13T^{2} \) |
| 17 | \( 1 - 0.911T + 17T^{2} \) |
| 19 | \( 1 + 0.375T + 19T^{2} \) |
| 23 | \( 1 + 0.153T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 - 5.66T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 3.62T + 41T^{2} \) |
| 43 | \( 1 + 0.855T + 43T^{2} \) |
| 47 | \( 1 + 8.00T + 47T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 59 | \( 1 - 9.27T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 + 0.739T + 71T^{2} \) |
| 73 | \( 1 + 8.19T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 - 0.101T + 89T^{2} \) |
| 97 | \( 1 - 4.01T + 97T^{2} \) |
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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
−8.323082334966808012193369951410, −7.56736644782199162795820232767, −6.73495536718760827914883789491, −6.36726777821559744713262867858, −5.16220318600016970961408349363, −4.10410698534293828936972288985, −3.51490759081213547605565102781, −2.94316888933666644874609700143, −1.73762844614143891257788591078, −0.60072733068810213797099636593,
0.60072733068810213797099636593, 1.73762844614143891257788591078, 2.94316888933666644874609700143, 3.51490759081213547605565102781, 4.10410698534293828936972288985, 5.16220318600016970961408349363, 6.36726777821559744713262867858, 6.73495536718760827914883789491, 7.56736644782199162795820232767, 8.323082334966808012193369951410
