| L(s) = 1 | − 2.38·3-s + 2.26·5-s − 3.26·7-s + 2.67·9-s + 3.48·11-s − 0.0423·13-s − 5.39·15-s + 0.559·17-s − 1.62·19-s + 7.78·21-s − 4.81·23-s + 0.135·25-s + 0.782·27-s − 6.07·29-s + 6.33·31-s − 8.30·33-s − 7.40·35-s − 9.01·37-s + 0.100·39-s + 7.61·41-s + 1.05·43-s + 6.05·45-s − 2.17·47-s + 3.68·49-s − 1.33·51-s + 4.67·53-s + 7.90·55-s + ⋯ |
| L(s) = 1 | − 1.37·3-s + 1.01·5-s − 1.23·7-s + 0.890·9-s + 1.05·11-s − 0.0117·13-s − 1.39·15-s + 0.135·17-s − 0.372·19-s + 1.69·21-s − 1.00·23-s + 0.0271·25-s + 0.150·27-s − 1.12·29-s + 1.13·31-s − 1.44·33-s − 1.25·35-s − 1.48·37-s + 0.0161·39-s + 1.18·41-s + 0.161·43-s + 0.902·45-s − 0.317·47-s + 0.526·49-s − 0.186·51-s + 0.642·53-s + 1.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9830410974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9830410974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 619 | \( 1 + T \) |
| good | 3 | \( 1 + 2.38T + 3T^{2} \) |
| 5 | \( 1 - 2.26T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 - 3.48T + 11T^{2} \) |
| 13 | \( 1 + 0.0423T + 13T^{2} \) |
| 17 | \( 1 - 0.559T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 + 4.81T + 23T^{2} \) |
| 29 | \( 1 + 6.07T + 29T^{2} \) |
| 31 | \( 1 - 6.33T + 31T^{2} \) |
| 37 | \( 1 + 9.01T + 37T^{2} \) |
| 41 | \( 1 - 7.61T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 + 2.17T + 47T^{2} \) |
| 53 | \( 1 - 4.67T + 53T^{2} \) |
| 59 | \( 1 - 6.68T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 4.11T + 71T^{2} \) |
| 73 | \( 1 + 6.53T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 6.39T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 2.73T + 97T^{2} \) |
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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
−7.34045273002344228420315848921, −6.59607647322747698089066520303, −6.28472259041342628174403352955, −5.79085793963464094418031135304, −5.20031966852223888432502452092, −4.20856386990369292244701242850, −3.57369179288717307246916507454, −2.47292783320216217194954253430, −1.56573841449335105903435286442, −0.51289166628414821874255318130,
0.51289166628414821874255318130, 1.56573841449335105903435286442, 2.47292783320216217194954253430, 3.57369179288717307246916507454, 4.20856386990369292244701242850, 5.20031966852223888432502452092, 5.79085793963464094418031135304, 6.28472259041342628174403352955, 6.59607647322747698089066520303, 7.34045273002344228420315848921
