| L(s) = 1 | − 2.29·2-s − 17.4·3-s − 26.7·4-s + 25·5-s + 40.0·6-s − 42.9·7-s + 134.·8-s + 61.1·9-s − 57.4·10-s − 121·11-s + 466.·12-s + 174.·13-s + 98.5·14-s − 436.·15-s + 545.·16-s − 2.22e3·17-s − 140.·18-s − 361·19-s − 668.·20-s + 748.·21-s + 277.·22-s + 406.·23-s − 2.35e3·24-s + 625·25-s − 400.·26-s + 3.17e3·27-s + 1.14e3·28-s + ⋯ |
| L(s) = 1 | − 0.405·2-s − 1.11·3-s − 0.835·4-s + 0.447·5-s + 0.454·6-s − 0.331·7-s + 0.744·8-s + 0.251·9-s − 0.181·10-s − 0.301·11-s + 0.934·12-s + 0.286·13-s + 0.134·14-s − 0.500·15-s + 0.532·16-s − 1.86·17-s − 0.102·18-s − 0.229·19-s − 0.373·20-s + 0.370·21-s + 0.122·22-s + 0.160·23-s − 0.833·24-s + 0.200·25-s − 0.116·26-s + 0.837·27-s + 0.276·28-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
| \(L(3)\) |
\(\approx\) |
\(0.2966249590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2966249590\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.29T + 32T^{2} \) |
| 3 | \( 1 + 17.4T + 243T^{2} \) |
| 7 | \( 1 + 42.9T + 1.68e4T^{2} \) |
| 13 | \( 1 - 174.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.22e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 406.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.08e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.67e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.78e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.25e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.29e4T + 8.58e9T^{2} \) |
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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
−9.070653854472226797355222165558, −8.690463878679771560558912451165, −7.48876117392685560405928983596, −6.46554570787000411390641947411, −5.86039568149716441091631179353, −4.86747510316278942801341697117, −4.27607777913078393479334252727, −2.79059997951998646682803974654, −1.41907473269842829713991416804, −0.28326593331393691579797179135,
0.28326593331393691579797179135, 1.41907473269842829713991416804, 2.79059997951998646682803974654, 4.27607777913078393479334252727, 4.86747510316278942801341697117, 5.86039568149716441091631179353, 6.46554570787000411390641947411, 7.48876117392685560405928983596, 8.690463878679771560558912451165, 9.070653854472226797355222165558
