| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s + 4·11-s + 12-s − 4·14-s + 16-s + 2·17-s + 18-s + 19-s − 4·21-s + 4·22-s + 2·23-s + 24-s + 27-s − 4·28-s − 6·29-s + 6·31-s + 32-s + 4·33-s + 2·34-s + 36-s + 8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.872·21-s + 0.852·22-s + 0.417·23-s + 0.204·24-s + 0.192·27-s − 0.755·28-s − 1.11·29-s + 1.07·31-s + 0.176·32-s + 0.696·33-s + 0.342·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.414798048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.414798048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.075576842406541201241594770774, −7.80954843090816149956704777250, −7.22822931232102684143493050402, −6.24265288481434745554412487281, −6.04547848262920215925763806415, −4.68475865669594320244755229333, −3.85809577170289018122041589134, −3.25425866639495471915764029739, −2.44562386899404817546775712870, −1.04269711977505524501535875684,
1.04269711977505524501535875684, 2.44562386899404817546775712870, 3.25425866639495471915764029739, 3.85809577170289018122041589134, 4.68475865669594320244755229333, 6.04547848262920215925763806415, 6.24265288481434745554412487281, 7.22822931232102684143493050402, 7.80954843090816149956704777250, 9.075576842406541201241594770774
