| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 4·11-s + 6·13-s + 15-s + 6·17-s − 4·19-s + 21-s + 4·23-s + 25-s − 27-s + 2·29-s − 8·31-s + 4·33-s + 35-s − 6·37-s − 6·39-s + 6·41-s + 8·43-s − 45-s + 49-s − 6·51-s − 6·53-s + 4·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.169·35-s − 0.986·37-s − 0.960·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.539·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.209130472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209130472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.79109574555346791256801405222, −7.43386623980937235365317958127, −6.44278460593601672528489601181, −5.85858678198854103520442806218, −5.28311978718186268276058419262, −4.38654278670578428538019687834, −3.55738793139165655919215071776, −2.94584336477056650141047357562, −1.63801114720454161830913248864, −0.59611944990633356231352825460,
0.59611944990633356231352825460, 1.63801114720454161830913248864, 2.94584336477056650141047357562, 3.55738793139165655919215071776, 4.38654278670578428538019687834, 5.28311978718186268276058419262, 5.85858678198854103520442806218, 6.44278460593601672528489601181, 7.43386623980937235365317958127, 7.79109574555346791256801405222
