| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 4·7-s + 9-s + 2·12-s + 5·13-s + 8·14-s − 4·16-s + 6·17-s + 2·18-s − 2·19-s + 4·21-s + 23-s + 10·26-s + 27-s + 8·28-s − 6·29-s − 8·31-s − 8·32-s + 12·34-s + 2·36-s + 2·37-s − 4·38-s + 5·39-s − 2·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1.51·7-s + 1/3·9-s + 0.577·12-s + 1.38·13-s + 2.13·14-s − 16-s + 1.45·17-s + 0.471·18-s − 0.458·19-s + 0.872·21-s + 0.208·23-s + 1.96·26-s + 0.192·27-s + 1.51·28-s − 1.11·29-s − 1.43·31-s − 1.41·32-s + 2.05·34-s + 1/3·36-s + 0.328·37-s − 0.648·38-s + 0.800·39-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.225198386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.225198386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.535786058608161691388036459475, −7.75482669549076536417944256340, −7.08052917253927092360893376776, −5.95272652473372080724672442893, −5.46872673780711415253216181133, −4.71013733038055956259223486220, −3.83342657973947117163068179491, −3.41865115482848661517391555018, −2.19802180792397448027125289341, −1.35525292470673817872478711742,
1.35525292470673817872478711742, 2.19802180792397448027125289341, 3.41865115482848661517391555018, 3.83342657973947117163068179491, 4.71013733038055956259223486220, 5.46872673780711415253216181133, 5.95272652473372080724672442893, 7.08052917253927092360893376776, 7.75482669549076536417944256340, 8.535786058608161691388036459475
