L(s) = 1 | + 1.77·3-s + 0.716·5-s + 0.155·9-s + 1.69·11-s + 4.41·13-s + 1.27·15-s − 1.35·17-s − 6.61·19-s − 4.94·23-s − 4.48·25-s − 5.05·27-s − 0.710·29-s − 3.41·31-s + 3.01·33-s − 5.21·37-s + 7.84·39-s + 41-s − 6.26·43-s + 0.111·45-s − 7.29·47-s − 2.41·51-s − 12.1·53-s + 1.21·55-s − 11.7·57-s + 1.69·59-s − 7.76·61-s + 3.16·65-s + ⋯ |
L(s) = 1 | + 1.02·3-s + 0.320·5-s + 0.0518·9-s + 0.511·11-s + 1.22·13-s + 0.328·15-s − 0.329·17-s − 1.51·19-s − 1.03·23-s − 0.897·25-s − 0.972·27-s − 0.132·29-s − 0.613·31-s + 0.524·33-s − 0.857·37-s + 1.25·39-s + 0.156·41-s − 0.955·43-s + 0.0166·45-s − 1.06·47-s − 0.337·51-s − 1.67·53-s + 0.164·55-s − 1.55·57-s + 0.220·59-s − 0.994·61-s + 0.392·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 1.77T + 3T^{2} \) |
| 5 | \( 1 - 0.716T + 5T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 + 4.94T + 23T^{2} \) |
| 29 | \( 1 + 0.710T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 + 7.29T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 1.69T + 59T^{2} \) |
| 61 | \( 1 + 7.76T + 61T^{2} \) |
| 67 | \( 1 - 3.13T + 67T^{2} \) |
| 71 | \( 1 - 3.54T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 8.90T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 6.08T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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
−7.83872252243791613255536237231, −6.59575544908390457657789560821, −6.29697440145161448780416090202, −5.50029820207739062251433260405, −4.42667142574961416898260277566, −3.75170241182274733509575256275, −3.21977394660452022343458188485, −2.04660342155584578883461340957, −1.71103912401574346651784818085, 0,
1.71103912401574346651784818085, 2.04660342155584578883461340957, 3.21977394660452022343458188485, 3.75170241182274733509575256275, 4.42667142574961416898260277566, 5.50029820207739062251433260405, 6.29697440145161448780416090202, 6.59575544908390457657789560821, 7.83872252243791613255536237231