| L(s) = 1 | + 0.397·2-s − 1.84·4-s + 3.16·5-s − 1.67·7-s − 1.52·8-s + 1.26·10-s − 1.35·11-s − 0.657·13-s − 0.665·14-s + 3.07·16-s − 4.41·17-s − 0.226·19-s − 5.83·20-s − 0.537·22-s + 23-s + 5.04·25-s − 0.261·26-s + 3.08·28-s − 29-s + 9.64·31-s + 4.27·32-s − 1.75·34-s − 5.30·35-s + 2.51·37-s − 0.0901·38-s − 4.84·40-s + 7.69·41-s + ⋯ |
| L(s) = 1 | + 0.281·2-s − 0.920·4-s + 1.41·5-s − 0.632·7-s − 0.539·8-s + 0.398·10-s − 0.407·11-s − 0.182·13-s − 0.177·14-s + 0.769·16-s − 1.07·17-s − 0.0520·19-s − 1.30·20-s − 0.114·22-s + 0.208·23-s + 1.00·25-s − 0.0512·26-s + 0.582·28-s − 0.185·29-s + 1.73·31-s + 0.756·32-s − 0.301·34-s − 0.897·35-s + 0.414·37-s − 0.0146·38-s − 0.765·40-s + 1.20·41-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 | \( 1 - 0.397T + 2T^{2} \) |
| 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 + 0.657T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 0.226T + 19T^{2} \) |
| 31 | \( 1 - 9.64T + 31T^{2} \) |
| 37 | \( 1 - 2.51T + 37T^{2} \) |
| 41 | \( 1 - 7.69T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 + 1.67T + 47T^{2} \) |
| 53 | \( 1 + 7.39T + 53T^{2} \) |
| 59 | \( 1 + 6.61T + 59T^{2} \) |
| 61 | \( 1 - 0.912T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 3.28T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 9.17T + 97T^{2} \) |
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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
−7.80845000742113392840306014391, −6.76774067223582221598925992796, −6.12045260531105414879027465705, −5.68874877784502803949909688689, −4.76225565152659819696571413225, −4.30357274142758009496825255263, −3.06156925881272105095090261451, −2.52385395241149472144848906083, −1.33567683756951260382506296095, 0,
1.33567683756951260382506296095, 2.52385395241149472144848906083, 3.06156925881272105095090261451, 4.30357274142758009496825255263, 4.76225565152659819696571413225, 5.68874877784502803949909688689, 6.12045260531105414879027465705, 6.76774067223582221598925992796, 7.80845000742113392840306014391
