| L(s) = 1 | − 5.02·2-s + 3.76·3-s − 6.70·4-s + 25·5-s − 18.9·6-s + 170.·7-s + 194.·8-s − 228.·9-s − 125.·10-s + 121·11-s − 25.2·12-s − 468.·13-s − 855.·14-s + 94.0·15-s − 764.·16-s + 646.·17-s + 1.15e3·18-s + 361·19-s − 167.·20-s + 639.·21-s − 608.·22-s + 2.56e3·23-s + 732.·24-s + 625·25-s + 2.35e3·26-s − 1.77e3·27-s − 1.14e3·28-s + ⋯ |
| L(s) = 1 | − 0.889·2-s + 0.241·3-s − 0.209·4-s + 0.447·5-s − 0.214·6-s + 1.31·7-s + 1.07·8-s − 0.941·9-s − 0.397·10-s + 0.301·11-s − 0.0505·12-s − 0.768·13-s − 1.16·14-s + 0.107·15-s − 0.746·16-s + 0.542·17-s + 0.837·18-s + 0.229·19-s − 0.0937·20-s + 0.316·21-s − 0.268·22-s + 1.00·23-s + 0.259·24-s + 0.200·25-s + 0.682·26-s − 0.468·27-s − 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.587871316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587871316\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 5.02T + 32T^{2} \) |
| 3 | \( 1 - 3.76T + 243T^{2} \) |
| 7 | \( 1 - 170.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 468.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 646.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 257.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.80e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.31e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.32e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.79e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.60e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.09e3T + 8.58e9T^{2} \) |
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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
−9.244291021476141589934830033532, −8.392779763592184981988525249817, −7.81473139947577486378532059710, −7.03862981124973427251654825921, −5.55124576648532111928107256263, −5.05423133072831057537782877822, −3.93892935502362020682509943321, −2.55142189572468678041826636571, −1.60521410687306818772764693284, −0.64943607206998097484964796504,
0.64943607206998097484964796504, 1.60521410687306818772764693284, 2.55142189572468678041826636571, 3.93892935502362020682509943321, 5.05423133072831057537782877822, 5.55124576648532111928107256263, 7.03862981124973427251654825921, 7.81473139947577486378532059710, 8.392779763592184981988525249817, 9.244291021476141589934830033532
