| L(s) = 1 | − 1.35·2-s − 0.150·4-s + 4.20·5-s + 0.870·7-s + 2.92·8-s − 5.71·10-s − 3.24·11-s − 1.60·13-s − 1.18·14-s − 3.67·16-s − 7.96·17-s − 1.89·19-s − 0.632·20-s + 4.40·22-s + 23-s + 12.6·25-s + 2.17·26-s − 0.131·28-s + 29-s − 6.87·31-s − 0.850·32-s + 10.8·34-s + 3.65·35-s + 5.65·37-s + 2.57·38-s + 12.2·40-s + 5.17·41-s + ⋯ |
| L(s) = 1 | − 0.961·2-s − 0.0753·4-s + 1.87·5-s + 0.328·7-s + 1.03·8-s − 1.80·10-s − 0.977·11-s − 0.444·13-s − 0.316·14-s − 0.918·16-s − 1.93·17-s − 0.434·19-s − 0.141·20-s + 0.940·22-s + 0.208·23-s + 2.52·25-s + 0.427·26-s − 0.0247·28-s + 0.185·29-s − 1.23·31-s − 0.150·32-s + 1.85·34-s + 0.617·35-s + 0.929·37-s + 0.417·38-s + 1.94·40-s + 0.807·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 5 | \( 1 - 4.20T + 5T^{2} \) |
| 7 | \( 1 - 0.870T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 + 7.96T + 17T^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 31 | \( 1 + 6.87T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 + 6.69T + 53T^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 + 0.0645T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 + 3.28T + 79T^{2} \) |
| 83 | \( 1 + 2.68T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−7.77686676053488193170718066469, −7.18717936564815332703700014059, −6.29616148505411027294667830819, −5.68803422783645702863347468513, −4.82966851032642784787673112373, −4.37861601224262722313147140622, −2.66074373342866920017279402354, −2.19008934239021513189002799799, −1.35700629050175259863854802370, 0,
1.35700629050175259863854802370, 2.19008934239021513189002799799, 2.66074373342866920017279402354, 4.37861601224262722313147140622, 4.82966851032642784787673112373, 5.68803422783645702863347468513, 6.29616148505411027294667830819, 7.18717936564815332703700014059, 7.77686676053488193170718066469
