| L(s) = 1 | − 3.29·3-s − 5-s − 3.51·7-s + 7.84·9-s + 4.27·13-s + 3.29·15-s − 1.58·17-s − 5.55·19-s + 11.5·21-s − 6.39·23-s + 25-s − 15.9·27-s + 0.381·29-s − 0.760·31-s + 3.51·35-s − 1.45·37-s − 14.0·39-s + 6.12·41-s + 7.19·43-s − 7.84·45-s + 4.32·47-s + 5.35·49-s + 5.23·51-s − 6.94·53-s + 18.2·57-s + 12.7·59-s + 8.89·61-s + ⋯ |
| L(s) = 1 | − 1.90·3-s − 0.447·5-s − 1.32·7-s + 2.61·9-s + 1.18·13-s + 0.850·15-s − 0.385·17-s − 1.27·19-s + 2.52·21-s − 1.33·23-s + 0.200·25-s − 3.06·27-s + 0.0707·29-s − 0.136·31-s + 0.594·35-s − 0.238·37-s − 2.25·39-s + 0.956·41-s + 1.09·43-s − 1.16·45-s + 0.630·47-s + 0.765·49-s + 0.733·51-s − 0.954·53-s + 2.42·57-s + 1.65·59-s + 1.13·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 3.29T + 3T^{2} \) |
| 7 | \( 1 + 3.51T + 7T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 + 5.55T + 19T^{2} \) |
| 23 | \( 1 + 6.39T + 23T^{2} \) |
| 29 | \( 1 - 0.381T + 29T^{2} \) |
| 31 | \( 1 + 0.760T + 31T^{2} \) |
| 37 | \( 1 + 1.45T + 37T^{2} \) |
| 41 | \( 1 - 6.12T + 41T^{2} \) |
| 43 | \( 1 - 7.19T + 43T^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 + 6.94T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 8.89T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 7.35T + 71T^{2} \) |
| 73 | \( 1 + 7.03T + 73T^{2} \) |
| 79 | \( 1 - 6.99T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 0.451T + 89T^{2} \) |
| 97 | \( 1 + 2.77T + 97T^{2} \) |
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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
−7.71993896385217450032126333318, −6.80147900488233766837529125579, −6.35463411134800628697082646903, −5.96421293741079258891260495846, −5.12409822430554533632348415164, −3.95346150043281398385445368796, −3.91242940979112008240273345846, −2.25097942854323415561040938841, −0.885940779843202223085647412953, 0,
0.885940779843202223085647412953, 2.25097942854323415561040938841, 3.91242940979112008240273345846, 3.95346150043281398385445368796, 5.12409822430554533632348415164, 5.96421293741079258891260495846, 6.35463411134800628697082646903, 6.80147900488233766837529125579, 7.71993896385217450032126333318
