| L(s) = 1 | + 2.77·5-s + 1.71·7-s − 0.585·11-s − 6.56·13-s − 0.228·17-s − 5.57·19-s − 4.63·23-s + 2.67·25-s + 6.25·29-s + 7.04·31-s + 4.74·35-s − 11.6·37-s − 2.26·41-s − 3.41·43-s + 6.83·47-s − 4.06·49-s − 9.69·53-s − 1.62·55-s − 9.29·59-s − 2.23·61-s − 18.1·65-s − 5.40·67-s + 10.7·71-s + 14.0·73-s − 1.00·77-s − 9.29·79-s − 14.0·83-s + ⋯ |
| L(s) = 1 | + 1.23·5-s + 0.647·7-s − 0.176·11-s − 1.82·13-s − 0.0553·17-s − 1.27·19-s − 0.966·23-s + 0.535·25-s + 1.16·29-s + 1.26·31-s + 0.802·35-s − 1.92·37-s − 0.354·41-s − 0.520·43-s + 0.997·47-s − 0.580·49-s − 1.33·53-s − 0.218·55-s − 1.21·59-s − 0.286·61-s − 2.25·65-s − 0.660·67-s + 1.27·71-s + 1.64·73-s − 0.114·77-s − 1.04·79-s − 1.54·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 2.77T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 + 0.585T + 11T^{2} \) |
| 13 | \( 1 + 6.56T + 13T^{2} \) |
| 17 | \( 1 + 0.228T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 - 6.25T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 + 3.41T + 43T^{2} \) |
| 47 | \( 1 - 6.83T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 + 9.29T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 + 5.40T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 9.29T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 2.45T + 89T^{2} \) |
| 97 | \( 1 - 4.76T + 97T^{2} \) |
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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
−7.86237531993066075464993694205, −6.83515027154361587661360917699, −6.39583921001115594105092723580, −5.48937414375433066417928983920, −4.88380204058465941862818373311, −4.31888551530560550169654969309, −2.94459815402758536067438093740, −2.21849668295077873425536057509, −1.61708716739617398299809786594, 0,
1.61708716739617398299809786594, 2.21849668295077873425536057509, 2.94459815402758536067438093740, 4.31888551530560550169654969309, 4.88380204058465941862818373311, 5.48937414375433066417928983920, 6.39583921001115594105092723580, 6.83515027154361587661360917699, 7.86237531993066075464993694205
