| L(s) = 1 | + 3-s + 2·7-s + 9-s + 4·11-s − 6·13-s + 6·17-s + 2·21-s + 4·23-s − 5·25-s + 27-s − 4·29-s + 10·31-s + 4·33-s − 2·37-s − 6·39-s − 2·41-s − 8·43-s − 12·47-s − 3·49-s + 6·51-s + 12·53-s + 4·59-s − 2·61-s + 2·63-s − 4·67-s + 4·69-s − 4·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.436·21-s + 0.834·23-s − 25-s + 0.192·27-s − 0.742·29-s + 1.79·31-s + 0.696·33-s − 0.328·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s − 1.75·47-s − 3/7·49-s + 0.840·51-s + 1.64·53-s + 0.520·59-s − 0.256·61-s + 0.251·63-s − 0.488·67-s + 0.481·69-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.789565998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.789565998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−11.69179794771208099164944938022, −10.12070860408780938514099192974, −9.634186835262726325802863954007, −8.503024042291174545390595385787, −7.67554775307177371631314255080, −6.80910210319960057293306406499, −5.36858848597467135489376840463, −4.36074490911347519116057298154, −3.06807634409715093203408940942, −1.59108885184574423909298130831,
1.59108885184574423909298130831, 3.06807634409715093203408940942, 4.36074490911347519116057298154, 5.36858848597467135489376840463, 6.80910210319960057293306406499, 7.67554775307177371631314255080, 8.503024042291174545390595385787, 9.634186835262726325802863954007, 10.12070860408780938514099192974, 11.69179794771208099164944938022
