| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 3·7-s + 8-s + 9-s − 10-s − 12-s + 13-s + 3·14-s + 15-s + 16-s + 2·17-s + 18-s + 19-s − 20-s − 3·21-s + 4·23-s − 24-s + 25-s + 26-s − 27-s + 3·28-s − 8·29-s + 30-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 − 0.288·12-s + 0.277·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.654·21-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.566·28-s − 1.48·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.75402676371252, −13.08539902373360, −12.73890815149320, −12.24759046781804, −11.70791205427830, −11.33785096848903, −10.99851817235969, −10.62508335345430, −9.948167228492300, −9.327486559045281, −8.794940641580238, −8.178904729734307, −7.590143297616466, −7.389660715894313, −6.772748812521695, −5.983933739667088, −5.745532870231176, −5.035628703640957, −4.698133129814762, −4.231410708938726, −3.445973520509746, −3.136056128070048, −2.137864169183762, −1.574500738127472, −1.010760605765422, 0,
1.010760605765422, 1.574500738127472, 2.137864169183762, 3.136056128070048, 3.445973520509746, 4.231410708938726, 4.698133129814762, 5.035628703640957, 5.745532870231176, 5.983933739667088, 6.772748812521695, 7.389660715894313, 7.590143297616466, 8.178904729734307, 8.794940641580238, 9.327486559045281, 9.948167228492300, 10.62508335345430, 10.99851817235969, 11.33785096848903, 11.70791205427830, 12.24759046781804, 12.73890815149320, 13.08539902373360, 13.75402676371252
