L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 2·7-s − 2·9-s + 2·12-s + 4·13-s + 4·14-s − 4·16-s − 2·17-s + 4·18-s − 2·21-s + 23-s − 8·26-s − 5·27-s − 4·28-s + 7·31-s + 8·32-s + 4·34-s − 4·36-s − 3·37-s + 4·39-s + 8·41-s + 4·42-s − 6·43-s − 2·46-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 0.755·7-s − 2/3·9-s + 0.577·12-s + 1.10·13-s + 1.06·14-s − 16-s − 0.485·17-s + 0.942·18-s − 0.436·21-s + 0.208·23-s − 1.56·26-s − 0.962·27-s − 0.755·28-s + 1.25·31-s + 1.41·32-s + 0.685·34-s − 2/3·36-s − 0.493·37-s + 0.640·39-s + 1.24·41-s + 0.617·42-s − 0.914·43-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7868271699\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7868271699\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.674636904258113471049328473744, −8.296604429873515576851811786575, −7.52659502080279327607409472268, −6.59063232546813848502583144427, −6.08680366637470864805213224746, −4.82789446851349902347971503063, −3.69673219139295712447524938606, −2.87981766717816431171166593097, −1.88537943180961640741129629846, −0.64215384115869760023600034446,
0.64215384115869760023600034446, 1.88537943180961640741129629846, 2.87981766717816431171166593097, 3.69673219139295712447524938606, 4.82789446851349902347971503063, 6.08680366637470864805213224746, 6.59063232546813848502583144427, 7.52659502080279327607409472268, 8.296604429873515576851811786575, 8.674636904258113471049328473744