| L(s) = 1 | + 1.10·2-s − 2.38·3-s − 0.770·4-s + 5-s − 2.64·6-s + 4.25·7-s − 3.07·8-s + 2.67·9-s + 1.10·10-s − 11-s + 1.83·12-s + 6.99·13-s + 4.71·14-s − 2.38·15-s − 1.86·16-s + 4.73·17-s + 2.96·18-s − 1.81·19-s − 0.770·20-s − 10.1·21-s − 1.10·22-s + 0.707·23-s + 7.32·24-s + 25-s + 7.75·26-s + 0.767·27-s − 3.27·28-s + ⋯ |
| L(s) = 1 | + 0.784·2-s − 1.37·3-s − 0.385·4-s + 0.447·5-s − 1.07·6-s + 1.60·7-s − 1.08·8-s + 0.892·9-s + 0.350·10-s − 0.301·11-s + 0.529·12-s + 1.93·13-s + 1.26·14-s − 0.615·15-s − 0.466·16-s + 1.14·17-s + 0.699·18-s − 0.415·19-s − 0.172·20-s − 2.21·21-s − 0.236·22-s + 0.147·23-s + 1.49·24-s + 0.200·25-s + 1.52·26-s + 0.147·27-s − 0.619·28-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
| \(L(1)\) |
\(\approx\) |
\(2.076185755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.076185755\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.10T + 2T^{2} \) |
| 3 | \( 1 + 2.38T + 3T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 13 | \( 1 - 6.99T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 - 0.707T + 23T^{2} \) |
| 29 | \( 1 + 0.895T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 79 | \( 1 - 4.11T + 79T^{2} \) |
| 83 | \( 1 - 6.33T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 3.71T + 97T^{2} \) |
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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
−8.528245583849704683659354043318, −7.69951330916354144474125510615, −6.53564147045220779514627055795, −5.96209963764364472947309386965, −5.33472074408333415171604507362, −4.97953589340045099295820286627, −4.12867167846169631194885881136, −3.26686320821059943583705082885, −1.72653047954976739603563142366, −0.853231638406285243731789789684,
0.853231638406285243731789789684, 1.72653047954976739603563142366, 3.26686320821059943583705082885, 4.12867167846169631194885881136, 4.97953589340045099295820286627, 5.33472074408333415171604507362, 5.96209963764364472947309386965, 6.53564147045220779514627055795, 7.69951330916354144474125510615, 8.528245583849704683659354043318
