| L(s) = 1 | + 0.609·2-s − 7.62·4-s + 11.6·5-s − 26.3·7-s − 9.53·8-s + 7.08·10-s + 0.610·11-s + 42.5·13-s − 16.0·14-s + 55.2·16-s − 4.31·17-s + 84.0·19-s − 88.5·20-s + 0.372·22-s − 97.3·23-s + 9.85·25-s + 25.9·26-s + 201.·28-s − 80.7·29-s − 155.·31-s + 109.·32-s − 2.63·34-s − 306.·35-s − 181.·37-s + 51.2·38-s − 110.·40-s + 77.9·41-s + ⋯ |
| L(s) = 1 | + 0.215·2-s − 0.953·4-s + 1.03·5-s − 1.42·7-s − 0.421·8-s + 0.223·10-s + 0.0167·11-s + 0.908·13-s − 0.307·14-s + 0.862·16-s − 0.0616·17-s + 1.01·19-s − 0.990·20-s + 0.00361·22-s − 0.882·23-s + 0.0788·25-s + 0.195·26-s + 1.35·28-s − 0.517·29-s − 0.898·31-s + 0.607·32-s − 0.0132·34-s − 1.47·35-s − 0.805·37-s + 0.218·38-s − 0.437·40-s + 0.297·41-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 | \( 1 - 0.609T + 8T^{2} \) |
| 5 | \( 1 - 11.6T + 125T^{2} \) |
| 7 | \( 1 + 26.3T + 343T^{2} \) |
| 11 | \( 1 - 0.610T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.31T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 80.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 77.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 481.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 215.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 321.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 464.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 392.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 179.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 216.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 49.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 561.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 750.T + 9.12e5T^{2} \) |
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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
−8.627696019128677869445570675945, −7.51489691252239884991178521412, −6.56542946740296753982543731430, −5.69562017875286481317501446479, −5.53009823872425021243904374565, −4.05063809625180925537812782417, −3.52873282778255540799914808102, −2.46559752342305405760893676889, −1.14529313351297300662219670624, 0,
1.14529313351297300662219670624, 2.46559752342305405760893676889, 3.52873282778255540799914808102, 4.05063809625180925537812782417, 5.53009823872425021243904374565, 5.69562017875286481317501446479, 6.56542946740296753982543731430, 7.51489691252239884991178521412, 8.627696019128677869445570675945
