| L(s) = 1 | − 2·2-s − 8.74·3-s + 4·4-s + 14.1·5-s + 17.4·6-s − 28.6·7-s − 8·8-s + 49.4·9-s − 28.3·10-s − 9.49·11-s − 34.9·12-s + 57.3·14-s − 123.·15-s + 16·16-s + 30.6·17-s − 98.8·18-s + 153.·19-s + 56.6·20-s + 250.·21-s + 18.9·22-s − 36.0·23-s + 69.9·24-s + 75.7·25-s − 195.·27-s − 114.·28-s − 49.2·29-s + 247.·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.68·3-s + 0.5·4-s + 1.26·5-s + 1.18·6-s − 1.54·7-s − 0.353·8-s + 1.82·9-s − 0.896·10-s − 0.260·11-s − 0.841·12-s + 1.09·14-s − 2.13·15-s + 0.250·16-s + 0.436·17-s − 1.29·18-s + 1.85·19-s + 0.633·20-s + 2.60·21-s + 0.184·22-s − 0.326·23-s + 0.594·24-s + 0.605·25-s − 1.39·27-s − 0.773·28-s − 0.315·29-s + 1.50·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 8.74T + 27T^{2} \) |
| 5 | \( 1 - 14.1T + 125T^{2} \) |
| 7 | \( 1 + 28.6T + 343T^{2} \) |
| 11 | \( 1 + 9.49T + 1.33e3T^{2} \) |
| 17 | \( 1 - 30.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 153.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 36.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 49.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 23.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 434.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 400.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 408.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 603.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 287.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 963.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 313.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 675.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 272.T + 9.12e5T^{2} \) |
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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
−10.33486292913899436312384140668, −9.925702154443383169015997146298, −9.314968194200030264335050396064, −7.50834841728014522185643938619, −6.44938967469256768990783514218, −6.00036715608835709021637542600, −5.11453507943084730085066258499, −3.09799543880049557304905635076, −1.31893702792134027444339857185, 0,
1.31893702792134027444339857185, 3.09799543880049557304905635076, 5.11453507943084730085066258499, 6.00036715608835709021637542600, 6.44938967469256768990783514218, 7.50834841728014522185643938619, 9.314968194200030264335050396064, 9.925702154443383169015997146298, 10.33486292913899436312384140668
