| L(s) = 1 | − 0.682·2-s + 3-s − 1.53·4-s + 3.92·5-s − 0.682·6-s + 1.75·7-s + 2.41·8-s + 9-s − 2.68·10-s − 1.87·11-s − 1.53·12-s − 2.72·13-s − 1.19·14-s + 3.92·15-s + 1.42·16-s − 1.56·17-s − 0.682·18-s + 0.433·19-s − 6.02·20-s + 1.75·21-s + 1.28·22-s + 3.96·23-s + 2.41·24-s + 10.4·25-s + 1.86·26-s + 27-s − 2.68·28-s + ⋯ |
| L(s) = 1 | − 0.482·2-s + 0.577·3-s − 0.766·4-s + 1.75·5-s − 0.278·6-s + 0.662·7-s + 0.853·8-s + 0.333·9-s − 0.847·10-s − 0.565·11-s − 0.442·12-s − 0.756·13-s − 0.319·14-s + 1.01·15-s + 0.355·16-s − 0.379·17-s − 0.160·18-s + 0.0994·19-s − 1.34·20-s + 0.382·21-s + 0.273·22-s + 0.826·23-s + 0.492·24-s + 2.08·25-s + 0.365·26-s + 0.192·27-s − 0.507·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484215610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484215610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.682T + 2T^{2} \) |
| 5 | \( 1 - 3.92T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 2.72T + 13T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 - 0.433T + 19T^{2} \) |
| 23 | \( 1 - 3.96T + 23T^{2} \) |
| 29 | \( 1 + 0.326T + 29T^{2} \) |
| 31 | \( 1 - 8.28T + 31T^{2} \) |
| 37 | \( 1 - 2.00T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 8.54T + 47T^{2} \) |
| 53 | \( 1 - 3.63T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 6.95T + 71T^{2} \) |
| 73 | \( 1 - 1.27T + 73T^{2} \) |
| 79 | \( 1 + 0.149T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−10.96979503052327292974631701168, −10.03832539591879613139536268640, −9.604802248774864943551838566290, −8.720646110308790552754341382614, −7.923550989486899289706019062258, −6.69145475018771112689878658956, −5.31872514558868582187557222151, −4.66938738103219722583571742864, −2.74237083628565218040139483912, −1.53327127188656880473129042260,
1.53327127188656880473129042260, 2.74237083628565218040139483912, 4.66938738103219722583571742864, 5.31872514558868582187557222151, 6.69145475018771112689878658956, 7.923550989486899289706019062258, 8.720646110308790552754341382614, 9.604802248774864943551838566290, 10.03832539591879613139536268640, 10.96979503052327292974631701168
