| L(s) = 1 | + 2·5-s − 3·9-s + 4·11-s + 2·13-s + 6·17-s + 8·19-s − 25-s − 6·29-s − 8·31-s + 2·37-s − 2·41-s + 4·43-s − 6·45-s + 8·47-s − 6·53-s + 8·55-s − 6·61-s + 4·65-s + 4·67-s − 8·71-s − 10·73-s + 16·79-s + 9·81-s + 8·83-s + 12·85-s + 6·89-s + 16·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.894·45-s + 1.16·47-s − 0.824·53-s + 1.07·55-s − 0.768·61-s + 0.496·65-s + 0.488·67-s − 0.949·71-s − 1.17·73-s + 1.80·79-s + 81-s + 0.878·83-s + 1.30·85-s + 0.635·89-s + 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.454753303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.454753303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.065821423802092395362181435296, −7.81893808816424475924755874673, −7.31356679414241138388846467495, −6.08808818165492598018676693090, −5.81038841174965326632002876390, −5.10093822894527537496894350174, −3.70152283742019411667382089257, −3.22234166202046004358507312699, −1.93975903006317803281774950018, −1.01536704926852438245703233735,
1.01536704926852438245703233735, 1.93975903006317803281774950018, 3.22234166202046004358507312699, 3.70152283742019411667382089257, 5.10093822894527537496894350174, 5.81038841174965326632002876390, 6.08808818165492598018676693090, 7.31356679414241138388846467495, 7.81893808816424475924755874673, 9.065821423802092395362181435296
