| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 7-s − 8-s + 9-s − 2·10-s + 12-s − 13-s + 14-s + 2·15-s + 16-s − 6·17-s − 18-s + 2·20-s − 21-s − 24-s − 25-s + 26-s + 27-s − 28-s + 2·29-s − 2·30-s − 32-s + 6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.447·20-s − 0.218·21-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.365·30-s − 0.176·32-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.89857299977917, −15.25217825059633, −14.74763741913205, −14.22391911370187, −13.48036598375155, −13.20934117895283, −12.69384762245278, −11.84819643285116, −11.37284315306500, −10.61639010125919, −10.18253707658320, −9.588397545455405, −9.198603604226136, −8.696100125210389, −8.065385216734113, −7.361188141426810, −6.854875654635310, −6.134875594719772, −5.788836577416461, −4.703928725605100, −4.191452695928546, −3.167431987930545, −2.517070112031965, −2.033913223019960, −1.157285474238917, 0,
1.157285474238917, 2.033913223019960, 2.517070112031965, 3.167431987930545, 4.191452695928546, 4.703928725605100, 5.788836577416461, 6.134875594719772, 6.854875654635310, 7.361188141426810, 8.065385216734113, 8.696100125210389, 9.198603604226136, 9.588397545455405, 10.18253707658320, 10.61639010125919, 11.37284315306500, 11.84819643285116, 12.69384762245278, 13.20934117895283, 13.48036598375155, 14.22391911370187, 14.74763741913205, 15.25217825059633, 15.89857299977917
