| L(s) = 1 | − 2.96·2-s + 3·3-s + 0.767·4-s − 5·5-s − 8.88·6-s − 25.4·7-s + 21.4·8-s + 9·9-s + 14.8·10-s + 2.30·12-s − 53.7·13-s + 75.3·14-s − 15·15-s − 69.5·16-s − 109.·17-s − 26.6·18-s + 96.4·19-s − 3.83·20-s − 76.3·21-s + 167.·23-s + 64.2·24-s + 25·25-s + 159.·26-s + 27·27-s − 19.5·28-s + 237.·29-s + 44.4·30-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.577·3-s + 0.0959·4-s − 0.447·5-s − 0.604·6-s − 1.37·7-s + 0.946·8-s + 0.333·9-s + 0.468·10-s + 0.0553·12-s − 1.14·13-s + 1.43·14-s − 0.258·15-s − 1.08·16-s − 1.56·17-s − 0.348·18-s + 1.16·19-s − 0.0428·20-s − 0.792·21-s + 1.51·23-s + 0.546·24-s + 0.200·25-s + 1.20·26-s + 0.192·27-s − 0.131·28-s + 1.51·29-s + 0.270·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.96T + 8T^{2} \) |
| 7 | \( 1 + 25.4T + 343T^{2} \) |
| 13 | \( 1 + 53.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 96.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 237.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4.59T + 2.97e4T^{2} \) |
| 37 | \( 1 + 36.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 204.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 143.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 160.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 93.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 228.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 832.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 695.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 252.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 931.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.37e3T + 9.12e5T^{2} \) |
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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
−8.730754622664679421829359189571, −7.83736128977553746251578683820, −7.05225367690550575003620643236, −6.65078089467648027974916133182, −5.08399392010695910448398905588, −4.32554128665068237326197299305, −3.19253810199639393198880819176, −2.43501563041991173238493202070, −0.940057136582848977585239592879, 0,
0.940057136582848977585239592879, 2.43501563041991173238493202070, 3.19253810199639393198880819176, 4.32554128665068237326197299305, 5.08399392010695910448398905588, 6.65078089467648027974916133182, 7.05225367690550575003620643236, 7.83736128977553746251578683820, 8.730754622664679421829359189571
