| L(s) = 1 | + 2·3-s − 2·4-s + 4·5-s + 2·7-s + 9-s − 11-s − 4·12-s + 8·15-s + 4·16-s + 2·17-s − 2·19-s − 8·20-s + 4·21-s + 4·23-s + 11·25-s − 4·27-s − 4·28-s + 8·29-s − 31-s − 2·33-s + 8·35-s − 2·36-s + 3·37-s − 4·41-s − 4·43-s + 2·44-s + 4·45-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 1.78·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.15·12-s + 2.06·15-s + 16-s + 0.485·17-s − 0.458·19-s − 1.78·20-s + 0.872·21-s + 0.834·23-s + 11/5·25-s − 0.769·27-s − 0.755·28-s + 1.48·29-s − 0.179·31-s − 0.348·33-s + 1.35·35-s − 1/3·36-s + 0.493·37-s − 0.624·41-s − 0.609·43-s + 0.301·44-s + 0.596·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.796822257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.796822257\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.361355527001060538698154807228, −7.83673108947455271038594427926, −6.73407814675951772156170824583, −5.92451071088820209988405651104, −5.13273452833804943447617503033, −4.73560855523020312627225266060, −3.54556581847717673895347912667, −2.75229119386493366252341437717, −1.98807888132825077073729318015, −1.08467906684834814522218068612,
1.08467906684834814522218068612, 1.98807888132825077073729318015, 2.75229119386493366252341437717, 3.54556581847717673895347912667, 4.73560855523020312627225266060, 5.13273452833804943447617503033, 5.92451071088820209988405651104, 6.73407814675951772156170824583, 7.83673108947455271038594427926, 8.361355527001060538698154807228
