| L(s) = 1 | + 5·5-s − 26.8·7-s − 46.6·11-s + 1.21·13-s − 87.2·17-s − 125.·19-s + 7.16·23-s + 25·25-s − 43.8·29-s + 140.·31-s − 134.·35-s − 187.·37-s + 239.·41-s + 457.·43-s + 25.6·47-s + 376.·49-s + 82.8·53-s − 233.·55-s − 739.·59-s − 28.6·61-s + 6.05·65-s − 126.·67-s + 611.·71-s + 983.·73-s + 1.25e3·77-s + 372.·79-s − 693.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.44·7-s − 1.27·11-s + 0.0258·13-s − 1.24·17-s − 1.51·19-s + 0.0649·23-s + 0.200·25-s − 0.280·29-s + 0.813·31-s − 0.647·35-s − 0.831·37-s + 0.913·41-s + 1.62·43-s + 0.0794·47-s + 1.09·49-s + 0.214·53-s − 0.571·55-s − 1.63·59-s − 0.0600·61-s + 0.0115·65-s − 0.230·67-s + 1.02·71-s + 1.57·73-s + 1.85·77-s + 0.530·79-s − 0.916·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8844272222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8844272222\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 26.8T + 343T^{2} \) |
| 11 | \( 1 + 46.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.21T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.16T + 1.21e4T^{2} \) |
| 29 | \( 1 + 43.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 187.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 239.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 457.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 25.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 82.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 739.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 28.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 126.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 611.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 983.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 372.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 693.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 873.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 82.7T + 9.12e5T^{2} \) |
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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
−9.121143993788365547165121508328, −8.345586933821936458320554870946, −7.33536442213888014818338721019, −6.45666109091250733751057136393, −6.00490437244136439881974239257, −4.90713799594990899045031791510, −3.96038143308473976700306512583, −2.80396059434008486668938981354, −2.18753538274157302000738077523, −0.42014222554540192525997178951,
0.42014222554540192525997178951, 2.18753538274157302000738077523, 2.80396059434008486668938981354, 3.96038143308473976700306512583, 4.90713799594990899045031791510, 6.00490437244136439881974239257, 6.45666109091250733751057136393, 7.33536442213888014818338721019, 8.345586933821936458320554870946, 9.121143993788365547165121508328
