| L(s) = 1 | + 1.56·2-s + 3-s + 0.438·4-s − 2·5-s + 1.56·6-s − 2.43·8-s + 9-s − 3.12·10-s + 6.56·11-s + 0.438·12-s + 3.12·13-s − 2·15-s − 4.68·16-s − 4.56·17-s + 1.56·18-s + 1.12·19-s − 0.876·20-s + 10.2·22-s − 2.43·24-s − 25-s + 4.87·26-s + 27-s + 5.68·29-s − 3.12·30-s − 5.43·31-s − 2.43·32-s + 6.56·33-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.577·3-s + 0.219·4-s − 0.894·5-s + 0.637·6-s − 0.862·8-s + 0.333·9-s − 0.987·10-s + 1.97·11-s + 0.126·12-s + 0.866·13-s − 0.516·15-s − 1.17·16-s − 1.10·17-s + 0.368·18-s + 0.257·19-s − 0.196·20-s + 2.18·22-s − 0.497·24-s − 0.200·25-s + 0.956·26-s + 0.192·27-s + 1.05·29-s − 0.570·30-s − 0.976·31-s − 0.431·32-s + 1.14·33-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
| \(L(1)\) |
\(\approx\) |
\(3.703915799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.703915799\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 6.56T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 5.43T + 31T^{2} \) |
| 37 | \( 1 - 9.68T + 37T^{2} \) |
| 43 | \( 1 + 0.315T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 0.315T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.131928472170879572856800513885, −7.22073987252636197244708812392, −6.46065620582856289235945884890, −6.08709395525927400775595693764, −4.88670174878828617795899535912, −4.17773838342022286915195821725, −3.83900963069345298776932357639, −3.22128901745501234479983521587, −2.09845582373673996782577839472, −0.852704754571029603161369632427,
0.852704754571029603161369632427, 2.09845582373673996782577839472, 3.22128901745501234479983521587, 3.83900963069345298776932357639, 4.17773838342022286915195821725, 4.88670174878828617795899535912, 6.08709395525927400775595693764, 6.46065620582856289235945884890, 7.22073987252636197244708812392, 8.131928472170879572856800513885
