| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 10.5·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s + 21.1·10-s + 68.6·11-s + 12·12-s − 43.1·13-s + 14·14-s − 31.6·15-s + 16·16-s − 47.5·17-s − 18·18-s − 19·19-s − 42.2·20-s − 21·21-s − 137.·22-s − 25.5·23-s − 24·24-s − 13.4·25-s + 86.3·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.944·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.668·10-s + 1.88·11-s + 0.288·12-s − 0.920·13-s + 0.267·14-s − 0.545·15-s + 0.250·16-s − 0.679·17-s − 0.235·18-s − 0.229·19-s − 0.472·20-s − 0.218·21-s − 1.33·22-s − 0.231·23-s − 0.204·24-s − 0.107·25-s + 0.651·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 + 10.5T + 125T^{2} \) |
| 11 | \( 1 - 68.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 25.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 336.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 144.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 411.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 493.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 335.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 848.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 811.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 540.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 245.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 288.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 281.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−9.538250820650580865640394962776, −8.476688111607030996430249016693, −8.032245710581676973035375838774, −6.79814352406052226583314388374, −6.52063741798776266819366598131, −4.64849633902018635454510636539, −3.81676345062926554541914246985, −2.75603354060513563819212554148, −1.39467399629265148750993824443, 0,
1.39467399629265148750993824443, 2.75603354060513563819212554148, 3.81676345062926554541914246985, 4.64849633902018635454510636539, 6.52063741798776266819366598131, 6.79814352406052226583314388374, 8.032245710581676973035375838774, 8.476688111607030996430249016693, 9.538250820650580865640394962776
