| L(s) = 1 | − 1.17·2-s − 3-s − 0.621·4-s + 3.86·5-s + 1.17·6-s + 3.06·7-s + 3.07·8-s + 9-s − 4.54·10-s + 1.94·11-s + 0.621·12-s − 4.31·13-s − 3.59·14-s − 3.86·15-s − 2.37·16-s − 17-s − 1.17·18-s − 5.08·19-s − 2.40·20-s − 3.06·21-s − 2.28·22-s − 2.65·23-s − 3.07·24-s + 9.96·25-s + 5.06·26-s − 27-s − 1.90·28-s + ⋯ |
| L(s) = 1 | − 0.830·2-s − 0.577·3-s − 0.310·4-s + 1.73·5-s + 0.479·6-s + 1.15·7-s + 1.08·8-s + 0.333·9-s − 1.43·10-s + 0.586·11-s + 0.179·12-s − 1.19·13-s − 0.960·14-s − 0.998·15-s − 0.592·16-s − 0.242·17-s − 0.276·18-s − 1.16·19-s − 0.537·20-s − 0.668·21-s − 0.487·22-s − 0.553·23-s − 0.628·24-s + 1.99·25-s + 0.992·26-s − 0.192·27-s − 0.359·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.404687219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404687219\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 + 4.31T + 13T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 - 5.36T + 29T^{2} \) |
| 31 | \( 1 - 0.729T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 - 8.81T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 - 5.95T + 47T^{2} \) |
| 53 | \( 1 + 1.03T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 5.20T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 83 | \( 1 - 1.61T + 83T^{2} \) |
| 89 | \( 1 + 2.37T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.740367831677847001318702007268, −7.79201615571536672117304088799, −7.01400555092410892018862665811, −6.23798458042492428231177148118, −5.45488400851186438701920103347, −4.79002357323332216443079056403, −4.22426447647384020320744069276, −2.34473055540631890587868020794, −1.82937221207146112348504451789, −0.832315915935347838629209976950,
0.832315915935347838629209976950, 1.82937221207146112348504451789, 2.34473055540631890587868020794, 4.22426447647384020320744069276, 4.79002357323332216443079056403, 5.45488400851186438701920103347, 6.23798458042492428231177148118, 7.01400555092410892018862665811, 7.79201615571536672117304088799, 8.740367831677847001318702007268
