| L(s) = 1 | + 1.76·2-s − 4.88·4-s + 16.9·5-s − 0.437·7-s − 22.7·8-s + 29.9·10-s + 26.6·11-s − 77.2·13-s − 0.771·14-s − 1.03·16-s + 23.3·17-s − 31.3·19-s − 83.0·20-s + 47.0·22-s + 81.5·23-s + 163.·25-s − 136.·26-s + 2.13·28-s + 42.1·29-s − 150.·31-s + 180.·32-s + 41.2·34-s − 7.42·35-s − 324.·37-s − 55.2·38-s − 386.·40-s + 3.36·41-s + ⋯ |
| L(s) = 1 | + 0.623·2-s − 0.610·4-s + 1.51·5-s − 0.0236·7-s − 1.00·8-s + 0.947·10-s + 0.730·11-s − 1.64·13-s − 0.0147·14-s − 0.0161·16-s + 0.333·17-s − 0.377·19-s − 0.927·20-s + 0.456·22-s + 0.739·23-s + 1.30·25-s − 1.02·26-s + 0.0144·28-s + 0.269·29-s − 0.872·31-s + 0.994·32-s + 0.207·34-s − 0.0358·35-s − 1.44·37-s − 0.235·38-s − 1.52·40-s + 0.0128·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 1.76T + 8T^{2} \) |
| 5 | \( 1 - 16.9T + 125T^{2} \) |
| 7 | \( 1 + 0.437T + 343T^{2} \) |
| 11 | \( 1 - 26.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 77.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 150.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 324.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 3.36T + 6.89e4T^{2} \) |
| 43 | \( 1 - 231.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 77.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 243.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 298.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 496.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 33.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 488.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 94.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 331.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 749.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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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
−8.625285371578396217603302636493, −7.36237246008833570248917347779, −6.57607784217575040175824291339, −5.77053939907309967166729175543, −5.14766079808160332282316089498, −4.49940104194206860035292576711, −3.33635271033656112067565928389, −2.44329227321880672637083799491, −1.42644094094431696151303218636, 0,
1.42644094094431696151303218636, 2.44329227321880672637083799491, 3.33635271033656112067565928389, 4.49940104194206860035292576711, 5.14766079808160332282316089498, 5.77053939907309967166729175543, 6.57607784217575040175824291339, 7.36237246008833570248917347779, 8.625285371578396217603302636493
