| L(s) = 1 | + 5.83·3-s + 12.0·5-s − 7·7-s + 7.02·9-s + 11·11-s + 31.7·13-s + 70.1·15-s + 112.·17-s − 6.19·19-s − 40.8·21-s + 113.·23-s + 19.5·25-s − 116.·27-s + 71.8·29-s − 88.9·31-s + 64.1·33-s − 84.1·35-s − 92.9·37-s + 184.·39-s + 192.·41-s + 124.·43-s + 84.4·45-s − 272.·47-s + 49·49-s + 653.·51-s − 122.·53-s + 132.·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s + 1.07·5-s − 0.377·7-s + 0.260·9-s + 0.301·11-s + 0.676·13-s + 1.20·15-s + 1.59·17-s − 0.0747·19-s − 0.424·21-s + 1.03·23-s + 0.156·25-s − 0.830·27-s + 0.459·29-s − 0.515·31-s + 0.338·33-s − 0.406·35-s − 0.412·37-s + 0.759·39-s + 0.733·41-s + 0.441·43-s + 0.279·45-s − 0.846·47-s + 0.142·49-s + 1.79·51-s − 0.316·53-s + 0.324·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.280980838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.280980838\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 5.83T + 27T^{2} \) |
| 5 | \( 1 - 12.0T + 125T^{2} \) |
| 13 | \( 1 - 31.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.19T + 6.85e3T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 71.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 92.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 192.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 124.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 67.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 173.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 22.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 769.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 57.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 177.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 243.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.81e3T + 9.12e5T^{2} \) |
| 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
−11.13546526351896884811350691564, −10.00722045034674589749914299720, −9.377141999511091939934182142648, −8.585963765913052979228992407085, −7.55668904252174798788164499014, −6.31102652166468182141009916993, −5.38150056405898238281279688524, −3.67195649218356068402521051489, −2.74045353411443523103340119925, −1.39631422895191903521809496927,
1.39631422895191903521809496927, 2.74045353411443523103340119925, 3.67195649218356068402521051489, 5.38150056405898238281279688524, 6.31102652166468182141009916993, 7.55668904252174798788164499014, 8.585963765913052979228992407085, 9.377141999511091939934182142648, 10.00722045034674589749914299720, 11.13546526351896884811350691564
