| L(s) = 1 | + 1.37·2-s + 1.23·3-s − 0.115·4-s + 1.69·6-s − 2.43·7-s − 2.90·8-s − 1.47·9-s − 11-s − 0.142·12-s + 4.84·13-s − 3.33·14-s − 3.75·16-s + 1.04·17-s − 2.01·18-s + 19-s − 3.00·21-s − 1.37·22-s − 0.377·23-s − 3.59·24-s + 6.65·26-s − 5.52·27-s + 0.280·28-s + 4.50·29-s + 4.76·31-s + 0.652·32-s − 1.23·33-s + 1.42·34-s + ⋯ |
| L(s) = 1 | + 0.970·2-s + 0.713·3-s − 0.0577·4-s + 0.692·6-s − 0.919·7-s − 1.02·8-s − 0.490·9-s − 0.301·11-s − 0.0412·12-s + 1.34·13-s − 0.892·14-s − 0.938·16-s + 0.252·17-s − 0.476·18-s + 0.229·19-s − 0.656·21-s − 0.292·22-s − 0.0786·23-s − 0.732·24-s + 1.30·26-s − 1.06·27-s + 0.0530·28-s + 0.836·29-s + 0.855·31-s + 0.115·32-s − 0.215·33-s + 0.244·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.914461667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.914461667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.37T + 2T^{2} \) |
| 3 | \( 1 - 1.23T + 3T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 23 | \( 1 + 0.377T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 - 4.76T + 31T^{2} \) |
| 37 | \( 1 + 3.01T + 37T^{2} \) |
| 41 | \( 1 - 0.790T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 1.32T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 - 1.20T + 67T^{2} \) |
| 71 | \( 1 + 0.259T + 71T^{2} \) |
| 73 | \( 1 - 6.27T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 1.23T + 89T^{2} \) |
| 97 | \( 1 + 2.95T + 97T^{2} \) |
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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
−8.400474448760444489680562618472, −7.47771111024756857991928763615, −6.47875614586230447299819915545, −5.96159146123688397159616512443, −5.36232833314700783684007596902, −4.31135697734167401105240837139, −3.64153213588919939963380827169, −3.07974878828912326409243293296, −2.39941133230605221911362720594, −0.75514681611091505281583979759,
0.75514681611091505281583979759, 2.39941133230605221911362720594, 3.07974878828912326409243293296, 3.64153213588919939963380827169, 4.31135697734167401105240837139, 5.36232833314700783684007596902, 5.96159146123688397159616512443, 6.47875614586230447299819915545, 7.47771111024756857991928763615, 8.400474448760444489680562618472
