| L(s) = 1 | + 1.73·2-s + 3·3-s − 4.98·4-s + 5.21·6-s + 7.66·7-s − 22.5·8-s + 9·9-s + 20.4·11-s − 14.9·12-s − 14.8·13-s + 13.3·14-s + 0.666·16-s − 108.·17-s + 15.6·18-s + 97.7·19-s + 22.9·21-s + 35.5·22-s − 80.5·23-s − 67.6·24-s − 25.7·26-s + 27·27-s − 38.1·28-s + 59.0·29-s + 176.·31-s + 181.·32-s + 61.4·33-s − 189.·34-s + ⋯ |
| L(s) = 1 | + 0.614·2-s + 0.577·3-s − 0.622·4-s + 0.354·6-s + 0.413·7-s − 0.996·8-s + 0.333·9-s + 0.561·11-s − 0.359·12-s − 0.315·13-s + 0.254·14-s + 0.0104·16-s − 1.55·17-s + 0.204·18-s + 1.18·19-s + 0.238·21-s + 0.344·22-s − 0.730·23-s − 0.575·24-s − 0.194·26-s + 0.192·27-s − 0.257·28-s + 0.378·29-s + 1.02·31-s + 1.00·32-s + 0.323·33-s − 0.955·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 8T^{2} \) |
| 7 | \( 1 - 7.66T + 343T^{2} \) |
| 11 | \( 1 - 20.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 97.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 80.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 59.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 230.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 477.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 515.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 571.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 52.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 551.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 368.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 406.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 654.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 421.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 425.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 590.T + 9.12e5T^{2} \) |
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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
−8.548951110796864841539842558470, −7.82021510087258198666151370450, −6.81291080888063094508083885944, −6.01728600637525739175700602203, −4.91866471613854161043348506504, −4.43085255278708569253826590120, −3.52672680124903082104861978757, −2.62455088865493123261445204678, −1.41691003589184874533265176849, 0,
1.41691003589184874533265176849, 2.62455088865493123261445204678, 3.52672680124903082104861978757, 4.43085255278708569253826590120, 4.91866471613854161043348506504, 6.01728600637525739175700602203, 6.81291080888063094508083885944, 7.82021510087258198666151370450, 8.548951110796864841539842558470
