L(s) = 1 | + 3-s + 2·5-s − 2.46·7-s − 2·9-s − 4·11-s + 2.46·13-s + 2·15-s + 5.92·17-s + 19-s − 2.46·21-s + 0.464·23-s − 25-s − 5·27-s − 4.46·29-s − 8.92·31-s − 4·33-s − 4.92·35-s + 2·37-s + 2.46·39-s + 6.92·41-s − 8.92·43-s − 4·45-s + 6.92·47-s − 0.928·49-s + 5.92·51-s − 5.53·53-s − 8·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.931·7-s − 0.666·9-s − 1.20·11-s + 0.683·13-s + 0.516·15-s + 1.43·17-s + 0.229·19-s − 0.537·21-s + 0.0967·23-s − 0.200·25-s − 0.962·27-s − 0.828·29-s − 1.60·31-s − 0.696·33-s − 0.833·35-s + 0.328·37-s + 0.394·39-s + 1.08·41-s − 1.36·43-s − 0.596·45-s + 1.01·47-s − 0.132·49-s + 0.830·51-s − 0.760·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 23 | \( 1 - 0.464T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 + 8.92T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 7.92T + 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 0.928T + 97T^{2} \) |
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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
−7.87684609255604018768593767162, −7.37635470433720034014778120929, −6.28536564355934383593988958156, −5.62924123225319106805395093059, −5.35746972936451176604603669929, −3.86510152142985138804017705955, −3.17904382845464435327691265716, −2.56962260051044799368333193724, −1.54972192335535122097301221981, 0,
1.54972192335535122097301221981, 2.56962260051044799368333193724, 3.17904382845464435327691265716, 3.86510152142985138804017705955, 5.35746972936451176604603669929, 5.62924123225319106805395093059, 6.28536564355934383593988958156, 7.37635470433720034014778120929, 7.87684609255604018768593767162