| L(s) = 1 | + 0.614·2-s − 9·3-s − 31.6·4-s − 56.4·5-s − 5.52·6-s − 140.·7-s − 39.0·8-s + 81·9-s − 34.7·10-s + 344.·11-s + 284.·12-s − 542.·13-s − 86.3·14-s + 508.·15-s + 987.·16-s + 1.20e3·17-s + 49.7·18-s − 1.13e3·19-s + 1.78e3·20-s + 1.26e3·21-s + 211.·22-s + 3.55e3·23-s + 351.·24-s + 65.7·25-s − 333.·26-s − 729·27-s + 4.44e3·28-s + ⋯ |
| L(s) = 1 | + 0.108·2-s − 0.577·3-s − 0.988·4-s − 1.01·5-s − 0.0627·6-s − 1.08·7-s − 0.215·8-s + 0.333·9-s − 0.109·10-s + 0.858·11-s + 0.570·12-s − 0.891·13-s − 0.117·14-s + 0.583·15-s + 0.964·16-s + 1.00·17-s + 0.0362·18-s − 0.723·19-s + 0.998·20-s + 0.626·21-s + 0.0932·22-s + 1.39·23-s + 0.124·24-s + 0.0210·25-s − 0.0967·26-s − 0.192·27-s + 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 - 0.614T + 32T^{2} \) |
| 5 | \( 1 + 56.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 140.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 344.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 542.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.13e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.77e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.22e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.65e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.59e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 706.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.97e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.68e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.99e4T + 8.58e9T^{2} \) |
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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
−9.676254234344578311567266752404, −9.059410405900337969935551235906, −7.904499886544357612215375709055, −6.98430809468752238678369573891, −5.97757079968421400423859743280, −4.84009891005398659464951774357, −4.00079409882882484527049114741, −3.10295149180665877940886980847, −0.907361944482416152730105879499, 0,
0.907361944482416152730105879499, 3.10295149180665877940886980847, 4.00079409882882484527049114741, 4.84009891005398659464951774357, 5.97757079968421400423859743280, 6.98430809468752238678369573891, 7.904499886544357612215375709055, 9.059410405900337969935551235906, 9.676254234344578311567266752404
