| L(s) = 1 | + 4·5-s + 20·11-s + 4·13-s + 24·17-s − 44·19-s − 72·23-s − 109·25-s + 38·29-s − 184·31-s − 30·37-s − 216·41-s − 164·43-s + 520·47-s + 146·53-s + 80·55-s + 460·59-s − 628·61-s + 16·65-s + 556·67-s − 592·71-s − 1.02e3·73-s − 104·79-s − 324·83-s + 96·85-s + 896·89-s − 176·95-s + 920·97-s + ⋯ |
| L(s) = 1 | + 0.357·5-s + 0.548·11-s + 0.0853·13-s + 0.342·17-s − 0.531·19-s − 0.652·23-s − 0.871·25-s + 0.243·29-s − 1.06·31-s − 0.133·37-s − 0.822·41-s − 0.581·43-s + 1.61·47-s + 0.378·53-s + 0.196·55-s + 1.01·59-s − 1.31·61-s + 0.0305·65-s + 1.01·67-s − 0.989·71-s − 1.64·73-s − 0.148·79-s − 0.428·83-s + 0.122·85-s + 1.06·89-s − 0.190·95-s + 0.963·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 T + p^{3} T^{2} \) |
| 17 | \( 1 - 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 38 T + p^{3} T^{2} \) |
| 31 | \( 1 + 184 T + p^{3} T^{2} \) |
| 37 | \( 1 + 30 T + p^{3} T^{2} \) |
| 41 | \( 1 + 216 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 520 T + p^{3} T^{2} \) |
| 53 | \( 1 - 146 T + p^{3} T^{2} \) |
| 59 | \( 1 - 460 T + p^{3} T^{2} \) |
| 61 | \( 1 + 628 T + p^{3} T^{2} \) |
| 67 | \( 1 - 556 T + p^{3} T^{2} \) |
| 71 | \( 1 + 592 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 79 | \( 1 + 104 T + p^{3} T^{2} \) |
| 83 | \( 1 + 324 T + p^{3} T^{2} \) |
| 89 | \( 1 - 896 T + p^{3} T^{2} \) |
| 97 | \( 1 - 920 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.631824740861872057556416940388, −7.75184053469148421510111225678, −6.92805008892479848421258759118, −6.07581117431351301630670690230, −5.41949830623708835622798278344, −4.29470056156461000616520711169, −3.53768921504853255726080105287, −2.31314883671426183931285282433, −1.39407604970780340297145861651, 0,
1.39407604970780340297145861651, 2.31314883671426183931285282433, 3.53768921504853255726080105287, 4.29470056156461000616520711169, 5.41949830623708835622798278344, 6.07581117431351301630670690230, 6.92805008892479848421258759118, 7.75184053469148421510111225678, 8.631824740861872057556416940388
