| L(s) = 1 | − 2.74·2-s + 5.53·4-s − 0.0839·5-s − 2.30·7-s − 9.69·8-s + 0.230·10-s − 4.15·11-s − 0.439·13-s + 6.31·14-s + 15.5·16-s + 0.388·17-s + 5.29·19-s − 0.464·20-s + 11.4·22-s + 23-s − 4.99·25-s + 1.20·26-s − 12.7·28-s − 29-s + 1.44·31-s − 23.2·32-s − 1.06·34-s + 0.193·35-s + 9.04·37-s − 14.5·38-s + 0.814·40-s + 0.159·41-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 2.76·4-s − 0.0375·5-s − 0.869·7-s − 3.42·8-s + 0.0728·10-s − 1.25·11-s − 0.121·13-s + 1.68·14-s + 3.88·16-s + 0.0941·17-s + 1.21·19-s − 0.103·20-s + 2.43·22-s + 0.208·23-s − 0.998·25-s + 0.236·26-s − 2.40·28-s − 0.185·29-s + 0.258·31-s − 4.11·32-s − 0.182·34-s + 0.0326·35-s + 1.48·37-s − 2.35·38-s + 0.128·40-s + 0.0248·41-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 | \( 1 + 2.74T + 2T^{2} \) |
| 5 | \( 1 + 0.0839T + 5T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 + 0.439T + 13T^{2} \) |
| 17 | \( 1 - 0.388T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 31 | \( 1 - 1.44T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 - 0.159T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 - 0.396T + 47T^{2} \) |
| 53 | \( 1 + 2.43T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 - 9.28T + 61T^{2} \) |
| 67 | \( 1 - 4.55T + 67T^{2} \) |
| 71 | \( 1 - 3.85T + 71T^{2} \) |
| 73 | \( 1 - 0.0113T + 73T^{2} \) |
| 79 | \( 1 - 9.57T + 79T^{2} \) |
| 83 | \( 1 - 5.55T + 83T^{2} \) |
| 89 | \( 1 - 4.02T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−7.933581117035656969101716950870, −7.28726973238515822076960300116, −6.61874725167955222645815994048, −5.89658764947822462615311328360, −5.16147940895514354099854802522, −3.59224061790230332320120049010, −2.84372850604105260583914244069, −2.17074186362400139921519542904, −0.958359351450519008807245281192, 0,
0.958359351450519008807245281192, 2.17074186362400139921519542904, 2.84372850604105260583914244069, 3.59224061790230332320120049010, 5.16147940895514354099854802522, 5.89658764947822462615311328360, 6.61874725167955222645815994048, 7.28726973238515822076960300116, 7.933581117035656969101716950870
