| 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 − 18-s − 3·20-s + 21-s + 4·22-s + 24-s + 4·25-s − 3·26-s − 27-s − 28-s + 29-s − 3·30-s − 2·31-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 − 0.235·18-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 + 0.185·29-s − 0.547·30-s − 0.359·31-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
| \(L(1)\) |
\(\approx\) |
\(0.4519408269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4519408269\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−15.80887059871237, −15.34891864958427, −14.61098645671540, −13.92717517086680, −13.08316714814769, −12.67590430056170, −12.27020756273973, −11.44161088256994, −11.05290826198115, −10.85345384533975, −9.989629110861990, −9.532088623237929, −8.737686056490956, −8.078558183090090, −7.825352921266932, −7.172111173070389, −6.545271112090857, −5.886122056295216, −5.245031553943053, −4.418319403649531, −3.807984146189689, −3.090955773748160, −2.338763245945859, −1.160043133588807, −0.3593021910023670,
0.3593021910023670, 1.160043133588807, 2.338763245945859, 3.090955773748160, 3.807984146189689, 4.418319403649531, 5.245031553943053, 5.886122056295216, 6.545271112090857, 7.172111173070389, 7.825352921266932, 8.078558183090090, 8.737686056490956, 9.532088623237929, 9.989629110861990, 10.85345384533975, 11.05290826198115, 11.44161088256994, 12.27020756273973, 12.67590430056170, 13.08316714814769, 13.92717517086680, 14.61098645671540, 15.34891864958427, 15.80887059871237
