| L(s) = 1 | − 3-s + 4·5-s + 9-s + 4·11-s + 4·13-s − 4·15-s + 4·19-s + 11·25-s − 27-s − 2·29-s − 8·31-s − 4·33-s + 6·37-s − 4·39-s − 4·43-s + 4·45-s + 8·47-s + 10·53-s + 16·55-s − 4·57-s + 4·59-s − 4·61-s + 16·65-s − 4·67-s + 8·71-s + 16·73-s − 11·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 1.03·15-s + 0.917·19-s + 11/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.640·39-s − 0.609·43-s + 0.596·45-s + 1.16·47-s + 1.37·53-s + 2.15·55-s − 0.529·57-s + 0.520·59-s − 0.512·61-s + 1.98·65-s − 0.488·67-s + 0.949·71-s + 1.87·73-s − 1.27·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.253429198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.253429198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.49420706310124187818607179085, −6.76855628347443180569354378677, −6.27797550792655846506930173140, −5.62337556861697863919174713587, −5.32665907333890564258931033533, −4.20903649814426964298715307886, −3.50924528680711880240892258370, −2.42517261412249507475649495972, −1.54874582990387817759591760844, −1.01084007980870082744625161268,
1.01084007980870082744625161268, 1.54874582990387817759591760844, 2.42517261412249507475649495972, 3.50924528680711880240892258370, 4.20903649814426964298715307886, 5.32665907333890564258931033533, 5.62337556861697863919174713587, 6.27797550792655846506930173140, 6.76855628347443180569354378677, 7.49420706310124187818607179085
