| L(s) = 1 | − 1.20·2-s + 0.184·3-s + 0.452·4-s − 0.222·6-s + 0.891·7-s + 0.659·8-s − 0.965·9-s + 0.0835·12-s − 1.07·14-s − 1.24·16-s + 1.47·17-s + 1.16·18-s + 1.86·19-s + 0.164·21-s − 1.70·23-s + 0.121·24-s + 25-s − 0.362·27-s + 0.403·28-s − 0.547·29-s + 1.47·31-s + 0.844·32-s − 1.78·34-s − 0.437·36-s − 2.24·38-s − 0.198·42-s − 1.96·43-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.184·3-s + 0.452·4-s − 0.222·6-s + 0.891·7-s + 0.659·8-s − 0.965·9-s + 0.0835·12-s − 1.07·14-s − 1.24·16-s + 1.47·17-s + 1.16·18-s + 1.86·19-s + 0.164·21-s − 1.70·23-s + 0.121·24-s + 25-s − 0.362·27-s + 0.403·28-s − 0.547·29-s + 1.47·31-s + 0.844·32-s − 1.78·34-s − 0.437·36-s − 2.24·38-s − 0.198·42-s − 1.96·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5058397731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5058397731\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 383 | \( 1 - T \) |
| good | 2 | \( 1 + 1.20T + T^{2} \) |
| 3 | \( 1 - 0.184T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.891T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.47T + T^{2} \) |
| 19 | \( 1 - 1.86T + T^{2} \) |
| 23 | \( 1 + 1.70T + T^{2} \) |
| 29 | \( 1 + 0.547T + T^{2} \) |
| 31 | \( 1 - 1.47T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.96T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.70T + T^{2} \) |
| 71 | \( 1 + 1.96T + T^{2} \) |
| 73 | \( 1 + 1.70T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−11.57404654250352235366322346112, −10.32945937183898458756537351103, −9.758400141855290309987027921255, −8.683983439354054797723730768459, −8.035945879104850228250535127044, −7.40626152370456132521394069717, −5.82320187234706372983611098365, −4.78449614205382577424308983650, −3.13829800942136174725975078055, −1.44306366962897073899379792133,
1.44306366962897073899379792133, 3.13829800942136174725975078055, 4.78449614205382577424308983650, 5.82320187234706372983611098365, 7.40626152370456132521394069717, 8.035945879104850228250535127044, 8.683983439354054797723730768459, 9.758400141855290309987027921255, 10.32945937183898458756537351103, 11.57404654250352235366322346112
