| L(s) = 1 | + 0.609·2-s + 3-s − 1.62·4-s + 2.30·5-s + 0.609·6-s − 1.80·7-s − 2.21·8-s + 9-s + 1.40·10-s − 0.459·11-s − 1.62·12-s − 13-s − 1.10·14-s + 2.30·15-s + 1.90·16-s − 4.35·17-s + 0.609·18-s − 5.50·19-s − 3.75·20-s − 1.80·21-s − 0.280·22-s − 0.178·23-s − 2.21·24-s + 0.320·25-s − 0.609·26-s + 27-s + 2.93·28-s + ⋯ |
| L(s) = 1 | + 0.431·2-s + 0.577·3-s − 0.814·4-s + 1.03·5-s + 0.248·6-s − 0.681·7-s − 0.782·8-s + 0.333·9-s + 0.444·10-s − 0.138·11-s − 0.470·12-s − 0.277·13-s − 0.294·14-s + 0.595·15-s + 0.476·16-s − 1.05·17-s + 0.143·18-s − 1.26·19-s − 0.839·20-s − 0.393·21-s − 0.0597·22-s − 0.0372·23-s − 0.451·24-s + 0.0641·25-s − 0.119·26-s + 0.192·27-s + 0.555·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.380021678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.380021678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.609T + 2T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 + 0.459T + 11T^{2} \) |
| 17 | \( 1 + 4.35T + 17T^{2} \) |
| 19 | \( 1 + 5.50T + 19T^{2} \) |
| 23 | \( 1 + 0.178T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 4.74T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 2.74T + 67T^{2} \) |
| 71 | \( 1 - 0.596T + 71T^{2} \) |
| 73 | \( 1 - 3.26T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 8.58T + 97T^{2} \) |
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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
−8.557623249363831862755302194730, −7.925044007422230109704099563472, −6.65193162733773539319973287955, −6.27602105598573923390095259688, −5.48051347289787242870673932504, −4.42046658627390426563425241311, −4.12025301662547862872824903253, −2.71201453573201137242381812573, −2.43400054016470623033401971831, −0.798597464063782447987728718048,
0.798597464063782447987728718048, 2.43400054016470623033401971831, 2.71201453573201137242381812573, 4.12025301662547862872824903253, 4.42046658627390426563425241311, 5.48051347289787242870673932504, 6.27602105598573923390095259688, 6.65193162733773539319973287955, 7.925044007422230109704099563472, 8.557623249363831862755302194730
