| L(s) = 1 | − 2.71·2-s + 5.37·4-s + 4.18·5-s + 1.30·7-s − 9.17·8-s − 11.3·10-s − 4.31·11-s + 3.47·13-s − 3.54·14-s + 14.1·16-s + 1.89·17-s + 3.72·19-s + 22.4·20-s + 11.7·22-s + 0.387·23-s + 12.4·25-s − 9.44·26-s + 7.01·28-s + 4.80·29-s − 10.0·31-s − 20.1·32-s − 5.14·34-s + 5.45·35-s − 8.33·37-s − 10.1·38-s − 38.3·40-s + 6.51·41-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.68·4-s + 1.86·5-s + 0.492·7-s − 3.24·8-s − 3.59·10-s − 1.30·11-s + 0.964·13-s − 0.946·14-s + 3.54·16-s + 0.459·17-s + 0.854·19-s + 5.02·20-s + 2.50·22-s + 0.0808·23-s + 2.49·25-s − 1.85·26-s + 1.32·28-s + 0.891·29-s − 1.81·31-s − 3.55·32-s − 0.882·34-s + 0.921·35-s − 1.37·37-s − 1.64·38-s − 6.06·40-s + 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.018295203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018295203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 + 4.31T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 - 3.72T + 19T^{2} \) |
| 23 | \( 1 - 0.387T + 23T^{2} \) |
| 29 | \( 1 - 4.80T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 8.33T + 37T^{2} \) |
| 41 | \( 1 - 6.51T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 0.536T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 1.22T + 59T^{2} \) |
| 61 | \( 1 + 0.628T + 61T^{2} \) |
| 67 | \( 1 + 0.0574T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 1.55T + 73T^{2} \) |
| 79 | \( 1 + 0.517T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 + 8.01T + 89T^{2} \) |
| 97 | \( 1 + 0.0296T + 97T^{2} \) |
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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
−10.01469299527328692861663424868, −9.219118996384323765353466008697, −8.674371714743640587071085167886, −7.72888486929238786454460269227, −6.94860037574607201740173831170, −5.84728845050179339997172390493, −5.42648818372594411275896226471, −2.97391036827163879836641505137, −2.05893355161221829071918061732, −1.14766489349425517995468970924,
1.14766489349425517995468970924, 2.05893355161221829071918061732, 2.97391036827163879836641505137, 5.42648818372594411275896226471, 5.84728845050179339997172390493, 6.94860037574607201740173831170, 7.72888486929238786454460269227, 8.674371714743640587071085167886, 9.219118996384323765353466008697, 10.01469299527328692861663424868
