| L(s) = 1 | − 2.35·2-s + 0.154·3-s + 3.53·4-s + 5-s − 0.363·6-s + 1.56·7-s − 3.60·8-s − 2.97·9-s − 2.35·10-s + 11-s + 0.546·12-s − 0.910·13-s − 3.69·14-s + 0.154·15-s + 1.41·16-s + 2.91·17-s + 7.00·18-s + 19-s + 3.53·20-s + 0.242·21-s − 2.35·22-s + 6.89·23-s − 0.557·24-s + 25-s + 2.14·26-s − 0.924·27-s + 5.54·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 0.0893·3-s + 1.76·4-s + 0.447·5-s − 0.148·6-s + 0.593·7-s − 1.27·8-s − 0.992·9-s − 0.743·10-s + 0.301·11-s + 0.157·12-s − 0.252·13-s − 0.986·14-s + 0.0399·15-s + 0.354·16-s + 0.705·17-s + 1.65·18-s + 0.229·19-s + 0.790·20-s + 0.0529·21-s − 0.501·22-s + 1.43·23-s − 0.113·24-s + 0.200·25-s + 0.419·26-s − 0.177·27-s + 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8226337521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8226337521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 0.154T + 3T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 13 | \( 1 + 0.910T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 - 3.33T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 + 0.359T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 2.31T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 67 | \( 1 - 5.61T + 67T^{2} \) |
| 71 | \( 1 - 8.92T + 71T^{2} \) |
| 73 | \( 1 - 7.09T + 73T^{2} \) |
| 79 | \( 1 + 5.34T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 3.19T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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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
−9.582378017625259820605387510624, −9.201297485749354887474866753892, −8.375636796378533494787476909267, −7.71486694632302373138359805468, −6.87838720182881119293280760244, −5.87311140210721598851472735041, −4.89488521951051958038330358561, −3.17687674207221823976764669789, −2.10568093143014271652104605496, −0.914673183198447386160518150988,
0.914673183198447386160518150988, 2.10568093143014271652104605496, 3.17687674207221823976764669789, 4.89488521951051958038330358561, 5.87311140210721598851472735041, 6.87838720182881119293280760244, 7.71486694632302373138359805468, 8.375636796378533494787476909267, 9.201297485749354887474866753892, 9.582378017625259820605387510624
