| L(s) = 1 | − 0.941·2-s + 1.53·3-s − 1.11·4-s − 5-s − 1.44·6-s + 7-s + 2.93·8-s − 0.628·9-s + 0.941·10-s − 1.71·12-s + 1.39·13-s − 0.941·14-s − 1.53·15-s − 0.533·16-s − 2.87·17-s + 0.591·18-s − 1.39·19-s + 1.11·20-s + 1.53·21-s + 6.46·23-s + 4.51·24-s + 25-s − 1.31·26-s − 5.58·27-s − 1.11·28-s − 2.72·29-s + 1.44·30-s + ⋯ |
| L(s) = 1 | − 0.665·2-s + 0.889·3-s − 0.556·4-s − 0.447·5-s − 0.591·6-s + 0.377·7-s + 1.03·8-s − 0.209·9-s + 0.297·10-s − 0.494·12-s + 0.385·13-s − 0.251·14-s − 0.397·15-s − 0.133·16-s − 0.697·17-s + 0.139·18-s − 0.319·19-s + 0.248·20-s + 0.336·21-s + 1.34·23-s + 0.921·24-s + 0.200·25-s − 0.256·26-s − 1.07·27-s − 0.210·28-s − 0.506·29-s + 0.264·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.941T + 2T^{2} \) |
| 3 | \( 1 - 1.53T + 3T^{2} \) |
| 13 | \( 1 - 1.39T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 23 | \( 1 - 6.46T + 23T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 + 2.10T + 31T^{2} \) |
| 37 | \( 1 + 6.54T + 37T^{2} \) |
| 41 | \( 1 + 7.54T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 - 8.87T + 47T^{2} \) |
| 53 | \( 1 - 7.31T + 53T^{2} \) |
| 59 | \( 1 + 8.87T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 - 0.00145T + 67T^{2} \) |
| 71 | \( 1 - 6.94T + 71T^{2} \) |
| 73 | \( 1 - 0.0957T + 73T^{2} \) |
| 79 | \( 1 + 9.24T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 0.516T + 89T^{2} \) |
| 97 | \( 1 - 0.474T + 97T^{2} \) |
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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
−8.324391645239516917215591696323, −7.48316565019553880015229478666, −7.00024861771781263219377589833, −5.73993423170798917046638867400, −4.92731712964669238710548611981, −4.10398746161757936347970932605, −3.41926448975827386826530331147, −2.37932799084057862861877294386, −1.33389879330898487803248102556, 0,
1.33389879330898487803248102556, 2.37932799084057862861877294386, 3.41926448975827386826530331147, 4.10398746161757936347970932605, 4.92731712964669238710548611981, 5.73993423170798917046638867400, 7.00024861771781263219377589833, 7.48316565019553880015229478666, 8.324391645239516917215591696323
