| L(s) = 1 | − 2.65·2-s − 2.98·3-s + 5.04·4-s + 5-s + 7.91·6-s − 3.81·7-s − 8.09·8-s + 5.88·9-s − 2.65·10-s − 5.97·11-s − 15.0·12-s + 2.24·13-s + 10.1·14-s − 2.98·15-s + 11.3·16-s − 1.94·17-s − 15.6·18-s + 0.247·19-s + 5.04·20-s + 11.3·21-s + 15.8·22-s + 2.92·23-s + 24.1·24-s + 25-s − 5.96·26-s − 8.58·27-s − 19.2·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 1.72·3-s + 2.52·4-s + 0.447·5-s + 3.23·6-s − 1.44·7-s − 2.86·8-s + 1.96·9-s − 0.839·10-s − 1.80·11-s − 4.34·12-s + 0.622·13-s + 2.70·14-s − 0.769·15-s + 2.84·16-s − 0.472·17-s − 3.68·18-s + 0.0567·19-s + 1.12·20-s + 2.48·21-s + 3.38·22-s + 0.610·23-s + 4.92·24-s + 0.200·25-s − 1.16·26-s − 1.65·27-s − 3.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2457197502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2457197502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 + 2.98T + 3T^{2} \) |
| 7 | \( 1 + 3.81T + 7T^{2} \) |
| 11 | \( 1 + 5.97T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 1.94T + 17T^{2} \) |
| 19 | \( 1 - 0.247T + 19T^{2} \) |
| 23 | \( 1 - 2.92T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 4.85T + 31T^{2} \) |
| 37 | \( 1 + 8.83T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 - 8.98T + 47T^{2} \) |
| 53 | \( 1 - 3.51T + 53T^{2} \) |
| 59 | \( 1 + 8.81T + 59T^{2} \) |
| 61 | \( 1 - 3.39T + 61T^{2} \) |
| 67 | \( 1 + 8.80T + 67T^{2} \) |
| 71 | \( 1 + 5.71T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 7.20T + 97T^{2} \) |
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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
−7.66290861375885519067289160074, −7.19619322179557429507956300191, −6.41786858781537890652651492771, −6.12382361288120593398789171791, −5.49823409145483063026300866349, −4.51971352711156405395148644681, −3.01090399251939029077039834159, −2.44912277171815982103206731889, −1.10072579926446595509127070378, −0.42851390798177252385424705293,
0.42851390798177252385424705293, 1.10072579926446595509127070378, 2.44912277171815982103206731889, 3.01090399251939029077039834159, 4.51971352711156405395148644681, 5.49823409145483063026300866349, 6.12382361288120593398789171791, 6.41786858781537890652651492771, 7.19619322179557429507956300191, 7.66290861375885519067289160074
