L(s) = 1 | + 2.41·2-s + 3.82·4-s + 5-s − 7-s + 4.41·8-s + 2.41·10-s + 0.414·11-s + 2.41·13-s − 2.41·14-s + 2.99·16-s + 4.82·17-s + 1.58·19-s + 3.82·20-s + 0.999·22-s − 0.828·23-s + 25-s + 5.82·26-s − 3.82·28-s − 4·29-s + 6·31-s − 1.58·32-s + 11.6·34-s − 35-s − 8.48·37-s + 3.82·38-s + 4.41·40-s − 7.82·41-s + ⋯ |
L(s) = 1 | + 1.70·2-s + 1.91·4-s + 0.447·5-s − 0.377·7-s + 1.56·8-s + 0.763·10-s + 0.124·11-s + 0.669·13-s − 0.645·14-s + 0.749·16-s + 1.17·17-s + 0.363·19-s + 0.856·20-s + 0.213·22-s − 0.172·23-s + 0.200·25-s + 1.14·26-s − 0.723·28-s − 0.742·29-s + 1.07·31-s − 0.280·32-s + 1.99·34-s − 0.169·35-s − 1.39·37-s + 0.621·38-s + 0.697·40-s − 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.491200233\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.491200233\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 + 6.65T + 43T^{2} \) |
| 47 | \( 1 - 7.48T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 6.48T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 7.82T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 2.07T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−10.26049845981630323212289996030, −9.368046954444422935177322869469, −8.199459174365219702779933570783, −7.08976445875095809099528335959, −6.30959450036543717872572883141, −5.61205151606384886952526799177, −4.83743794234285426491415549875, −3.66332065022488016517928301561, −3.05711879703359342182885052726, −1.67389539528351470154202294170,
1.67389539528351470154202294170, 3.05711879703359342182885052726, 3.66332065022488016517928301561, 4.83743794234285426491415549875, 5.61205151606384886952526799177, 6.30959450036543717872572883141, 7.08976445875095809099528335959, 8.199459174365219702779933570783, 9.368046954444422935177322869469, 10.26049845981630323212289996030