| L(s) = 1 | − 0.879·2-s − 1.22·4-s + 2.53·5-s − 1.87·7-s + 2.83·8-s − 2.22·10-s − 3.41·11-s − 5.29·13-s + 1.65·14-s − 0.0418·16-s − 1.65·17-s − 3.10·20-s + 2.99·22-s − 1.75·23-s + 1.41·25-s + 4.65·26-s + 2.30·28-s + 3.46·29-s + 1.94·31-s − 5.63·32-s + 1.45·34-s − 4.75·35-s − 0.837·37-s + 7.18·40-s + 4.49·41-s + 4.80·43-s + 4.18·44-s + ⋯ |
| L(s) = 1 | − 0.621·2-s − 0.613·4-s + 1.13·5-s − 0.710·7-s + 1.00·8-s − 0.704·10-s − 1.02·11-s − 1.46·13-s + 0.441·14-s − 0.0104·16-s − 0.400·17-s − 0.694·20-s + 0.639·22-s − 0.366·23-s + 0.282·25-s + 0.912·26-s + 0.435·28-s + 0.643·29-s + 0.349·31-s − 0.996·32-s + 0.249·34-s − 0.804·35-s − 0.137·37-s + 1.13·40-s + 0.701·41-s + 0.732·43-s + 0.630·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8360141293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8360141293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.879T + 2T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 + 0.837T + 37T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 - 4.80T + 43T^{2} \) |
| 47 | \( 1 + 0.716T + 47T^{2} \) |
| 53 | \( 1 - 6.10T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 7.51T + 73T^{2} \) |
| 79 | \( 1 + 6.96T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 - 2.28T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−8.779827273951064011120685903489, −8.006622685922770289135450936856, −7.27085506354603653711790197897, −6.46823620019014427882502897453, −5.49669965028132236366087015244, −5.00619892203851388773668521777, −4.04485276297994712151056046912, −2.72203125060311555762232778393, −2.08499135756128324225191113191, −0.58223475523779210953423923149,
0.58223475523779210953423923149, 2.08499135756128324225191113191, 2.72203125060311555762232778393, 4.04485276297994712151056046912, 5.00619892203851388773668521777, 5.49669965028132236366087015244, 6.46823620019014427882502897453, 7.27085506354603653711790197897, 8.006622685922770289135450936856, 8.779827273951064011120685903489
