| L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s − 4·11-s + 13-s + 2·15-s + 2·17-s + 4·21-s − 25-s + 27-s − 10·29-s − 4·31-s − 4·33-s + 8·35-s − 2·37-s + 39-s + 6·41-s + 12·43-s + 2·45-s + 9·49-s + 2·51-s + 6·53-s − 8·55-s − 12·59-s − 2·61-s + 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.516·15-s + 0.485·17-s + 0.872·21-s − 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.696·33-s + 1.35·35-s − 0.328·37-s + 0.160·39-s + 0.937·41-s + 1.82·43-s + 0.298·45-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 1.07·55-s − 1.56·59-s − 0.256·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.286588633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.286588633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.68841471949187060786371352877, −9.664804319908762705380195155746, −8.883561417319989083154185952335, −7.87562910104581673219715889473, −7.44185114017878719646522591958, −5.80299685653230421706138124069, −5.23507120944175378476623079671, −4.02605909040568767080832687408, −2.52449297382629447816032554487, −1.62121826810470584698639893491,
1.62121826810470584698639893491, 2.52449297382629447816032554487, 4.02605909040568767080832687408, 5.23507120944175378476623079671, 5.80299685653230421706138124069, 7.44185114017878719646522591958, 7.87562910104581673219715889473, 8.883561417319989083154185952335, 9.664804319908762705380195155746, 10.68841471949187060786371352877
