| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 3·14-s − 15-s + 16-s + 17-s + 18-s − 4·19-s + 20-s − 3·21-s − 22-s − 3·23-s − 24-s − 4·25-s − 26-s − 27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.654·21-s − 0.213·22-s − 0.625·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.35719331177610, −15.75353937029641, −15.11836063960568, −14.75841375506084, −14.00972167820127, −13.71826432457856, −13.00494344018932, −12.41184853682921, −11.86383285511576, −11.44087837209468, −10.78379754697715, −10.24097033368341, −9.795315395946547, −8.803602086045924, −8.082432538355835, −7.729216192554122, −6.772027690725815, −6.294721928100134, −5.646788861800541, −4.918329068840833, −4.657708897485278, −3.799441092980748, −2.870605376180626, −1.935010943496837, −1.477098351649472, 0,
1.477098351649472, 1.935010943496837, 2.870605376180626, 3.799441092980748, 4.657708897485278, 4.918329068840833, 5.646788861800541, 6.294721928100134, 6.772027690725815, 7.729216192554122, 8.082432538355835, 8.803602086045924, 9.795315395946547, 10.24097033368341, 10.78379754697715, 11.44087837209468, 11.86383285511576, 12.41184853682921, 13.00494344018932, 13.71826432457856, 14.00972167820127, 14.75841375506084, 15.11836063960568, 15.75353937029641, 16.35719331177610
