| L(s) = 1 | − 1.32·2-s + 3-s − 0.246·4-s + 4.03·5-s − 1.32·6-s + 2.97·8-s + 9-s − 5.34·10-s + 4.32·11-s − 0.246·12-s + 1.67·13-s + 4.03·15-s − 3.44·16-s − 1.53·17-s − 1.32·18-s + 4.63·19-s − 0.995·20-s − 5.72·22-s + 1.68·23-s + 2.97·24-s + 11.2·25-s − 2.22·26-s + 27-s + 6.19·29-s − 5.34·30-s − 7.82·31-s − 1.38·32-s + ⋯ |
| L(s) = 1 | − 0.936·2-s + 0.577·3-s − 0.123·4-s + 1.80·5-s − 0.540·6-s + 1.05·8-s + 0.333·9-s − 1.68·10-s + 1.30·11-s − 0.0712·12-s + 0.465·13-s + 1.04·15-s − 0.861·16-s − 0.372·17-s − 0.312·18-s + 1.06·19-s − 0.222·20-s − 1.21·22-s + 0.350·23-s + 0.607·24-s + 2.25·25-s − 0.435·26-s + 0.192·27-s + 1.14·29-s − 0.975·30-s − 1.40·31-s − 0.245·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.506109034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.506109034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 5 | \( 1 - 4.03T + 5T^{2} \) |
| 11 | \( 1 - 4.32T + 11T^{2} \) |
| 13 | \( 1 - 1.67T + 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 - 1.68T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + 7.82T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 + 7.25T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 - 6.75T + 59T^{2} \) |
| 61 | \( 1 + 1.19T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 + 5.99T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 + 0.0883T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 2.82T + 97T^{2} \) |
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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
−8.379485815431346758445804080859, −7.43766040012280070579372095064, −6.74946028228645232686984193866, −6.10965723469070294307326706153, −5.26393747790032848418764169923, −4.48165415012821944415885819100, −3.51187985278840866041913146501, −2.48928087312513173790389010865, −1.55966390217817631281731593454, −1.09352384669276846253458790048,
1.09352384669276846253458790048, 1.55966390217817631281731593454, 2.48928087312513173790389010865, 3.51187985278840866041913146501, 4.48165415012821944415885819100, 5.26393747790032848418764169923, 6.10965723469070294307326706153, 6.74946028228645232686984193866, 7.43766040012280070579372095064, 8.379485815431346758445804080859
