L(s) = 1 | + 0.456·2-s − 1.79·4-s − 1.73·8-s + 0.818·11-s + 2.79·16-s + 0.373·22-s − 5.29·23-s − 5·25-s + 10.5·29-s + 4.73·32-s + 6.16·37-s − 1.41·43-s − 1.46·44-s − 2.41·46-s − 2.28·50-s − 13.9·53-s + 4.83·58-s − 3.41·64-s − 15.7·67-s − 16.5·71-s + 2.81·74-s + 9.74·79-s − 0.647·86-s − 1.41·88-s + 9.47·92-s + 8.95·100-s − 6.37·106-s + ⋯ |
L(s) = 1 | + 0.323·2-s − 0.895·4-s − 0.612·8-s + 0.246·11-s + 0.697·16-s + 0.0797·22-s − 1.10·23-s − 25-s + 1.96·29-s + 0.837·32-s + 1.01·37-s − 0.216·43-s − 0.220·44-s − 0.356·46-s − 0.323·50-s − 1.91·53-s + 0.634·58-s − 0.427·64-s − 1.92·67-s − 1.95·71-s + 0.327·74-s + 1.09·79-s − 0.0698·86-s − 0.151·88-s + 0.988·92-s + 0.895·100-s − 0.619·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.456T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 0.818T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 9.74T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.142756292199330047939904185547, −7.51837709677910742511135768780, −6.28625730064740959887235522715, −5.98446153688585560844118214729, −4.87898586020754082598129614038, −4.37537709022199179604942745837, −3.55973404828077194567090508261, −2.65201577598597211162084295258, −1.33745529344424522133029828183, 0,
1.33745529344424522133029828183, 2.65201577598597211162084295258, 3.55973404828077194567090508261, 4.37537709022199179604942745837, 4.87898586020754082598129614038, 5.98446153688585560844118214729, 6.28625730064740959887235522715, 7.51837709677910742511135768780, 8.142756292199330047939904185547