| L(s) = 1 | + 5-s + 2·11-s + 2.82·13-s + 6.82·17-s + 2.58·19-s + 4.58·23-s + 25-s + 7.65·29-s − 4.24·31-s + 6.48·37-s + 2·41-s − 9.65·43-s − 6.48·47-s + 7.41·53-s + 2·55-s + 2.82·59-s + 9.89·61-s + 2.82·65-s − 1.17·67-s − 6.48·71-s − 12.4·73-s − 10·79-s − 0.828·83-s + 6.82·85-s − 0.828·89-s + 2.58·95-s + 4·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.603·11-s + 0.784·13-s + 1.65·17-s + 0.593·19-s + 0.956·23-s + 0.200·25-s + 1.42·29-s − 0.762·31-s + 1.06·37-s + 0.312·41-s − 1.47·43-s − 0.945·47-s + 1.01·53-s + 0.269·55-s + 0.368·59-s + 1.26·61-s + 0.350·65-s − 0.143·67-s − 0.769·71-s − 1.46·73-s − 1.12·79-s − 0.0909·83-s + 0.740·85-s − 0.0878·89-s + 0.265·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.001037057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.001037057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 - 7.41T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 0.828T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−7.74712033516730431473372574603, −7.01383579709068447954719237792, −6.38741445795664512481647015119, −5.65849528119130049581914814029, −5.13380425038798007123967452937, −4.19882927046724117591422500221, −3.36959714532770377512894494684, −2.79825987665496331429859113815, −1.51308334383486734356373123213, −0.952848583396612255265534872428,
0.952848583396612255265534872428, 1.51308334383486734356373123213, 2.79825987665496331429859113815, 3.36959714532770377512894494684, 4.19882927046724117591422500221, 5.13380425038798007123967452937, 5.65849528119130049581914814029, 6.38741445795664512481647015119, 7.01383579709068447954719237792, 7.74712033516730431473372574603
