| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 2·7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 3·13-s + 2·14-s − 15-s + 16-s − 6·17-s − 18-s − 20-s − 2·21-s − 22-s − 24-s − 4·25-s + 3·26-s + 27-s − 2·28-s − 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 − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.832·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.223·20-s − 0.436·21-s − 0.213·22-s − 0.204·24-s − 4/5·25-s + 0.588·26-s + 0.192·27-s − 0.377·28-s − 0.185·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−15.38606228749165, −15.10408678698035, −14.63036650093085, −13.79659452849288, −13.48541965749267, −12.66360422275017, −12.47032712219729, −11.65786871825626, −11.24820707174785, −10.63297469033998, −9.959446555994327, −9.579963278144313, −9.060975450078529, −8.533836243797047, −8.055778001395619, −7.219704627661723, −7.003586328420901, −6.444889465079115, −5.608266492332665, −4.882012663951550, −4.110767465990729, −3.491142092044125, −2.960066979702046, −2.033347327966164, −1.638216955237713, 0, 0,
1.638216955237713, 2.033347327966164, 2.960066979702046, 3.491142092044125, 4.110767465990729, 4.882012663951550, 5.608266492332665, 6.444889465079115, 7.003586328420901, 7.219704627661723, 8.055778001395619, 8.533836243797047, 9.060975450078529, 9.579963278144313, 9.959446555994327, 10.63297469033998, 11.24820707174785, 11.65786871825626, 12.47032712219729, 12.66360422275017, 13.48541965749267, 13.79659452849288, 14.63036650093085, 15.10408678698035, 15.38606228749165
