L(s) = 1 | + 0.487·2-s − 1.76·4-s + 2.22·5-s − 0.937·7-s − 1.83·8-s + 1.08·10-s + 2.32·11-s − 4.97·13-s − 0.456·14-s + 2.63·16-s + 7.85·17-s + 7.13·19-s − 3.92·20-s + 1.13·22-s − 6.70·23-s − 0.0401·25-s − 2.42·26-s + 1.65·28-s + 2.92·29-s + 3.83·31-s + 4.94·32-s + 3.82·34-s − 2.08·35-s + 5.27·37-s + 3.47·38-s − 4.08·40-s + 7.47·41-s + ⋯ |
L(s) = 1 | + 0.344·2-s − 0.881·4-s + 0.995·5-s − 0.354·7-s − 0.648·8-s + 0.343·10-s + 0.701·11-s − 1.38·13-s − 0.122·14-s + 0.658·16-s + 1.90·17-s + 1.63·19-s − 0.877·20-s + 0.241·22-s − 1.39·23-s − 0.00802·25-s − 0.475·26-s + 0.312·28-s + 0.543·29-s + 0.689·31-s + 0.874·32-s + 0.655·34-s − 0.352·35-s + 0.867·37-s + 0.564·38-s − 0.645·40-s + 1.16·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.832747609\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.832747609\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 - 0.487T + 2T^{2} \) |
| 5 | \( 1 - 2.22T + 5T^{2} \) |
| 7 | \( 1 + 0.937T + 7T^{2} \) |
| 11 | \( 1 - 2.32T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 - 7.85T + 17T^{2} \) |
| 19 | \( 1 - 7.13T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 6.09T + 47T^{2} \) |
| 53 | \( 1 - 5.27T + 53T^{2} \) |
| 59 | \( 1 - 0.126T + 59T^{2} \) |
| 61 | \( 1 - 0.511T + 61T^{2} \) |
| 67 | \( 1 - 6.08T + 67T^{2} \) |
| 71 | \( 1 + 9.50T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 0.569T + 79T^{2} \) |
| 83 | \( 1 + 8.63T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−9.716427047225782921074281302908, −9.444624683355057581965065299885, −8.118548796442977467350344365497, −7.40338367560061548235995208717, −6.07733339221565482135838977567, −5.61588942843343345139701699396, −4.71768347416768630919138133086, −3.63568818637942112659170288785, −2.63404172126485708835416752802, −1.03468672043055913295076328627,
1.03468672043055913295076328627, 2.63404172126485708835416752802, 3.63568818637942112659170288785, 4.71768347416768630919138133086, 5.61588942843343345139701699396, 6.07733339221565482135838977567, 7.40338367560061548235995208717, 8.118548796442977467350344365497, 9.444624683355057581965065299885, 9.716427047225782921074281302908