| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 7-s + 8-s + 9-s − 3·10-s − 4·11-s + 12-s − 3·13-s + 14-s − 3·15-s + 16-s + 4·17-s + 18-s − 3·20-s + 21-s − 4·22-s + 24-s + 4·25-s − 3·26-s + 27-s + 28-s + 3·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 1.20·11-s + 0.288·12-s − 0.832·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.670·20-s + 0.218·21-s − 0.852·22-s + 0.204·24-s + 4/5·25-s − 0.588·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.77714870282122, −15.01580329576621, −14.72671118993419, −14.44320152020856, −13.60349378468572, −13.01230618404069, −12.62201489286784, −12.00930700009114, −11.62166128951789, −10.90081734147937, −10.47942782225066, −9.765863288832910, −9.106444251967951, −8.206800378307825, −7.857690297568590, −7.493261345667970, −7.002447800358863, −5.918324273486683, −5.389289905128725, −4.566208059344880, −4.290627161660511, −3.428315290114449, −2.868913293153329, −2.284314103281577, −1.138724637365272, 0,
1.138724637365272, 2.284314103281577, 2.868913293153329, 3.428315290114449, 4.290627161660511, 4.566208059344880, 5.389289905128725, 5.918324273486683, 7.002447800358863, 7.493261345667970, 7.857690297568590, 8.206800378307825, 9.106444251967951, 9.765863288832910, 10.47942782225066, 10.90081734147937, 11.62166128951789, 12.00930700009114, 12.62201489286784, 13.01230618404069, 13.60349378468572, 14.44320152020856, 14.72671118993419, 15.01580329576621, 15.77714870282122
