| L(s) = 1 | + 2.50·2-s + 1.48·3-s + 4.25·4-s − 3.61·5-s + 3.71·6-s + 7-s + 5.64·8-s − 0.794·9-s − 9.04·10-s + 2.63·11-s + 6.32·12-s − 5.17·13-s + 2.50·14-s − 5.37·15-s + 5.60·16-s + 3.60·17-s − 1.98·18-s + 0.435·19-s − 15.3·20-s + 1.48·21-s + 6.58·22-s − 8.24·23-s + 8.38·24-s + 8.08·25-s − 12.9·26-s − 5.63·27-s + 4.25·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.857·3-s + 2.12·4-s − 1.61·5-s + 1.51·6-s + 0.377·7-s + 1.99·8-s − 0.264·9-s − 2.86·10-s + 0.793·11-s + 1.82·12-s − 1.43·13-s + 0.668·14-s − 1.38·15-s + 1.40·16-s + 0.873·17-s − 0.468·18-s + 0.0999·19-s − 3.44·20-s + 0.324·21-s + 1.40·22-s − 1.71·23-s + 1.71·24-s + 1.61·25-s − 2.53·26-s − 1.08·27-s + 0.804·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.404648147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.404648147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 + 5.17T + 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 - 0.435T + 19T^{2} \) |
| 23 | \( 1 + 8.24T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 - 2.69T + 41T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 - 5.00T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 5.06T + 67T^{2} \) |
| 71 | \( 1 - 7.48T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 3.80T + 79T^{2} \) |
| 83 | \( 1 + 2.34T + 83T^{2} \) |
| 89 | \( 1 - 2.32T + 89T^{2} \) |
| 97 | \( 1 - 5.49T + 97T^{2} \) |
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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
−11.97690919945210638680477605070, −11.49954656378280586594331177074, −10.09113593566685189266091998765, −8.493528048186404681304523462677, −7.70991615492046330926096706659, −6.85613282395556030533148490963, −5.38535031536638533774483775231, −4.26835415600470484502782691082, −3.60639557020504476968686782104, −2.50749028558571409601400623175,
2.50749028558571409601400623175, 3.60639557020504476968686782104, 4.26835415600470484502782691082, 5.38535031536638533774483775231, 6.85613282395556030533148490963, 7.70991615492046330926096706659, 8.493528048186404681304523462677, 10.09113593566685189266091998765, 11.49954656378280586594331177074, 11.97690919945210638680477605070
