| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·7-s + 8-s + 9-s − 3·11-s + 12-s + 4·13-s + 3·14-s + 16-s + 17-s + 18-s − 5·19-s + 3·21-s − 3·22-s + 4·23-s + 24-s + 4·26-s + 27-s + 3·28-s + 7·31-s + 32-s − 3·33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 1.10·13-s + 0.801·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.14·19-s + 0.654·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.566·28-s + 1.25·31-s + 0.176·32-s − 0.522·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.140253590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.140253590\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.486406270359151923545108175638, −8.284704267374673619707814716884, −7.44328836613652674818715542050, −6.52628311605829456535195406292, −5.69123633134174195032778666477, −4.80044701985141564292427048345, −4.21723005261880299588421682144, −3.16141119853209555134238453735, −2.30580017218184760862870268601, −1.26707773095087684500223038881,
1.26707773095087684500223038881, 2.30580017218184760862870268601, 3.16141119853209555134238453735, 4.21723005261880299588421682144, 4.80044701985141564292427048345, 5.69123633134174195032778666477, 6.52628311605829456535195406292, 7.44328836613652674818715542050, 8.284704267374673619707814716884, 8.486406270359151923545108175638
