| L(s) = 1 | + 0.470·2-s + 3-s − 1.77·4-s + 5-s + 0.470·6-s − 7-s − 1.77·8-s + 9-s + 0.470·10-s + 11-s − 1.77·12-s − 0.249·13-s − 0.470·14-s + 15-s + 2.71·16-s + 5.71·17-s + 0.470·18-s + 1.83·19-s − 1.77·20-s − 21-s + 0.470·22-s + 0.778·23-s − 1.77·24-s + 25-s − 0.117·26-s + 27-s + 1.77·28-s + ⋯ |
| L(s) = 1 | + 0.332·2-s + 0.577·3-s − 0.889·4-s + 0.447·5-s + 0.192·6-s − 0.377·7-s − 0.628·8-s + 0.333·9-s + 0.148·10-s + 0.301·11-s − 0.513·12-s − 0.0690·13-s − 0.125·14-s + 0.258·15-s + 0.679·16-s + 1.38·17-s + 0.110·18-s + 0.421·19-s − 0.397·20-s − 0.218·21-s + 0.100·22-s + 0.162·23-s − 0.363·24-s + 0.200·25-s − 0.0229·26-s + 0.192·27-s + 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.077294659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.077294659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 13 | \( 1 + 0.249T + 13T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 - 0.778T + 23T^{2} \) |
| 29 | \( 1 - 3.47T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 9.80T + 41T^{2} \) |
| 43 | \( 1 + 5.96T + 43T^{2} \) |
| 47 | \( 1 - 3.30T + 47T^{2} \) |
| 53 | \( 1 + 2.77T + 53T^{2} \) |
| 59 | \( 1 + 3.58T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 - 5.55T + 67T^{2} \) |
| 71 | \( 1 - 2.74T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 7.15T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−9.613132801103725993998897010878, −9.181494347527020757395188260251, −8.205201349979257110692197035165, −7.46908858325147721820875227971, −6.25876079029069407740083361424, −5.51321268685974033765276187276, −4.53132558662130568780621810578, −3.58181207372389250448723130375, −2.74827704447577099792691429509, −1.08756302667123128444735616533,
1.08756302667123128444735616533, 2.74827704447577099792691429509, 3.58181207372389250448723130375, 4.53132558662130568780621810578, 5.51321268685974033765276187276, 6.25876079029069407740083361424, 7.46908858325147721820875227971, 8.205201349979257110692197035165, 9.181494347527020757395188260251, 9.613132801103725993998897010878
