| L(s) = 1 | + 0.285·2-s + 1.26·3-s − 1.91·4-s + 0.362·6-s + 2.73·7-s − 1.11·8-s − 1.39·9-s + 3.99·11-s − 2.43·12-s − 6.64·13-s + 0.780·14-s + 3.51·16-s − 1.76·17-s − 0.397·18-s − 4.25·19-s + 3.46·21-s + 1.14·22-s + 7.54·23-s − 1.41·24-s − 1.89·26-s − 5.56·27-s − 5.24·28-s − 0.649·29-s − 9.61·31-s + 3.24·32-s + 5.06·33-s − 0.505·34-s + ⋯ |
| L(s) = 1 | + 0.201·2-s + 0.732·3-s − 0.959·4-s + 0.147·6-s + 1.03·7-s − 0.395·8-s − 0.463·9-s + 1.20·11-s − 0.702·12-s − 1.84·13-s + 0.208·14-s + 0.879·16-s − 0.429·17-s − 0.0936·18-s − 0.975·19-s + 0.756·21-s + 0.243·22-s + 1.57·23-s − 0.289·24-s − 0.372·26-s − 1.07·27-s − 0.991·28-s − 0.120·29-s − 1.72·31-s + 0.573·32-s + 0.882·33-s − 0.0866·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 197 | \( 1 + T \) |
| good | 2 | \( 1 - 0.285T + 2T^{2} \) |
| 3 | \( 1 - 1.26T + 3T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 3.99T + 11T^{2} \) |
| 13 | \( 1 + 6.64T + 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 + 4.25T + 19T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 + 0.649T + 29T^{2} \) |
| 31 | \( 1 + 9.61T + 31T^{2} \) |
| 37 | \( 1 - 9.93T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 - 7.01T + 43T^{2} \) |
| 47 | \( 1 + 5.51T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 7.62T + 67T^{2} \) |
| 71 | \( 1 + 2.65T + 71T^{2} \) |
| 73 | \( 1 + 5.54T + 73T^{2} \) |
| 79 | \( 1 - 7.00T + 79T^{2} \) |
| 83 | \( 1 - 0.0317T + 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.81533777089248382322540436225, −7.54505124349765572732250431984, −6.40857889434702740927868746660, −5.55074864522796048337123912836, −4.61845329821299021714600092581, −4.43156549682650425737774846430, −3.31882598783517194310065552290, −2.49895181303491012379432088317, −1.48629164949941588777256189126, 0,
1.48629164949941588777256189126, 2.49895181303491012379432088317, 3.31882598783517194310065552290, 4.43156549682650425737774846430, 4.61845329821299021714600092581, 5.55074864522796048337123912836, 6.40857889434702740927868746660, 7.54505124349765572732250431984, 7.81533777089248382322540436225
