| L(s) = 1 | − 2.62·2-s + 4.88·4-s − 0.233·5-s + 1.03·7-s − 7.57·8-s + 0.612·10-s + 11-s + 3.46·13-s − 2.71·14-s + 10.1·16-s − 0.866·17-s − 6.81·19-s − 1.14·20-s − 2.62·22-s + 2.06·23-s − 4.94·25-s − 9.08·26-s + 5.06·28-s + 6.06·29-s + 0.640·31-s − 11.3·32-s + 2.27·34-s − 0.241·35-s + 6.49·37-s + 17.8·38-s + 1.76·40-s + 0.461·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.44·4-s − 0.104·5-s + 0.391·7-s − 2.67·8-s + 0.193·10-s + 0.301·11-s + 0.960·13-s − 0.726·14-s + 2.52·16-s − 0.210·17-s − 1.56·19-s − 0.255·20-s − 0.559·22-s + 0.429·23-s − 0.989·25-s − 1.78·26-s + 0.956·28-s + 1.12·29-s + 0.115·31-s − 2.01·32-s + 0.389·34-s − 0.0408·35-s + 1.06·37-s + 2.90·38-s + 0.279·40-s + 0.0720·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 + 0.233T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 0.866T + 17T^{2} \) |
| 19 | \( 1 + 6.81T + 19T^{2} \) |
| 23 | \( 1 - 2.06T + 23T^{2} \) |
| 29 | \( 1 - 6.06T + 29T^{2} \) |
| 31 | \( 1 - 0.640T + 31T^{2} \) |
| 37 | \( 1 - 6.49T + 37T^{2} \) |
| 41 | \( 1 - 0.461T + 41T^{2} \) |
| 43 | \( 1 + 5.84T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.93T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 4.62T + 83T^{2} \) |
| 89 | \( 1 + 4.71T + 89T^{2} \) |
| 97 | \( 1 + 0.332T + 97T^{2} \) |
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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
−8.042009649216156021355255264130, −7.23820656974106375375347762432, −6.33308980450579895462553065170, −6.23764877333009453460378234354, −4.85343872269836339396968643383, −3.88122845138455707177053977899, −2.83331466444041189110011038894, −1.91829880081318649649063335747, −1.19359520793810345402548041058, 0,
1.19359520793810345402548041058, 1.91829880081318649649063335747, 2.83331466444041189110011038894, 3.88122845138455707177053977899, 4.85343872269836339396968643383, 6.23764877333009453460378234354, 6.33308980450579895462553065170, 7.23820656974106375375347762432, 8.042009649216156021355255264130
