| L(s) = 1 | − 2·2-s − 3.82·3-s + 4·4-s − 15.1·5-s + 7.65·6-s + 1.61·7-s − 8·8-s − 12.3·9-s + 30.3·10-s + 38.5·11-s − 15.3·12-s + 3.99·13-s − 3.22·14-s + 58.1·15-s + 16·16-s − 89.3·17-s + 24.7·18-s − 60.7·20-s − 6.17·21-s − 77.0·22-s + 67.5·23-s + 30.6·24-s + 105.·25-s − 7.99·26-s + 150.·27-s + 6.45·28-s + 266.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.736·3-s + 0.5·4-s − 1.35·5-s + 0.520·6-s + 0.0871·7-s − 0.353·8-s − 0.457·9-s + 0.961·10-s + 1.05·11-s − 0.368·12-s + 0.0852·13-s − 0.0615·14-s + 1.00·15-s + 0.250·16-s − 1.27·17-s + 0.323·18-s − 0.679·20-s − 0.0641·21-s − 0.746·22-s + 0.611·23-s + 0.260·24-s + 0.847·25-s − 0.0602·26-s + 1.07·27-s + 0.0435·28-s + 1.70·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 3.82T + 27T^{2} \) |
| 5 | \( 1 + 15.1T + 125T^{2} \) |
| 7 | \( 1 - 1.61T + 343T^{2} \) |
| 11 | \( 1 - 38.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.99T + 2.19e3T^{2} \) |
| 17 | \( 1 + 89.3T + 4.91e3T^{2} \) |
| 23 | \( 1 - 67.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 266.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 247.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 369.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 407.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 588.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 806.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 204.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 714.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 106.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 878.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 472.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.64e3T + 9.12e5T^{2} \) |
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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
−9.478937267639274215124134306852, −8.585025597999839647961641790795, −8.058260272698973339050051778003, −6.79678004098010322656590900146, −6.43475111144206041585550970056, −4.96422486522512393642941800890, −4.04975587415080738437704833126, −2.81009550970670346712551918980, −1.04294085680259568894438475402, 0,
1.04294085680259568894438475402, 2.81009550970670346712551918980, 4.04975587415080738437704833126, 4.96422486522512393642941800890, 6.43475111144206041585550970056, 6.79678004098010322656590900146, 8.058260272698973339050051778003, 8.585025597999839647961641790795, 9.478937267639274215124134306852
