| L(s) = 1 | − 2.76·2-s + 1.77·3-s + 5.67·4-s + 5-s − 4.91·6-s − 0.953·7-s − 10.1·8-s + 0.154·9-s − 2.76·10-s − 11-s + 10.0·12-s − 6.96·13-s + 2.64·14-s + 1.77·15-s + 16.8·16-s − 1.81·17-s − 0.427·18-s − 0.165·19-s + 5.67·20-s − 1.69·21-s + 2.76·22-s + 1.66·23-s − 18.0·24-s + 25-s + 19.3·26-s − 5.05·27-s − 5.41·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 1.02·3-s + 2.83·4-s + 0.447·5-s − 2.00·6-s − 0.360·7-s − 3.59·8-s + 0.0514·9-s − 0.875·10-s − 0.301·11-s + 2.90·12-s − 1.93·13-s + 0.706·14-s + 0.458·15-s + 4.20·16-s − 0.441·17-s − 0.100·18-s − 0.0380·19-s + 1.26·20-s − 0.369·21-s + 0.590·22-s + 0.347·23-s − 3.68·24-s + 0.200·25-s + 3.78·26-s − 0.972·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8554590572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8554590572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 - 1.77T + 3T^{2} \) |
| 7 | \( 1 + 0.953T + 7T^{2} \) |
| 13 | \( 1 + 6.96T + 13T^{2} \) |
| 17 | \( 1 + 1.81T + 17T^{2} \) |
| 19 | \( 1 + 0.165T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 - 7.43T + 29T^{2} \) |
| 31 | \( 1 - 5.38T + 31T^{2} \) |
| 37 | \( 1 + 1.22T + 37T^{2} \) |
| 41 | \( 1 - 9.23T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 - 3.29T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 - 2.86T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 5.28T + 67T^{2} \) |
| 71 | \( 1 + 2.20T + 71T^{2} \) |
| 79 | \( 1 + 7.82T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 9.62T + 89T^{2} \) |
| 97 | \( 1 + 3.04T + 97T^{2} \) |
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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
−8.621918817420932755041832929811, −7.78087370330429321651449274225, −7.45252772420440866323577564890, −6.60841427786899610987617687035, −5.90432580190701501262187671325, −4.70272576996773193128361513686, −3.08876883075572183615214102999, −2.60517708452588012887085372169, −2.04058254360693353908430317336, −0.63624238114930055613344122732,
0.63624238114930055613344122732, 2.04058254360693353908430317336, 2.60517708452588012887085372169, 3.08876883075572183615214102999, 4.70272576996773193128361513686, 5.90432580190701501262187671325, 6.60841427786899610987617687035, 7.45252772420440866323577564890, 7.78087370330429321651449274225, 8.621918817420932755041832929811
