L(s) = 1 | + 2·5-s + 2·7-s + 11-s − 4·13-s − 17-s + 2·19-s + 2·23-s − 25-s + 2·29-s + 4·31-s + 4·35-s − 6·37-s − 6·41-s + 2·43-s − 3·49-s − 12·53-s + 2·55-s + 14·59-s − 6·61-s − 8·65-s − 4·67-s − 2·71-s − 8·73-s + 2·77-s + 2·79-s + 12·83-s − 2·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.301·11-s − 1.10·13-s − 0.242·17-s + 0.458·19-s + 0.417·23-s − 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.676·35-s − 0.986·37-s − 0.937·41-s + 0.304·43-s − 3/7·49-s − 1.64·53-s + 0.269·55-s + 1.82·59-s − 0.768·61-s − 0.992·65-s − 0.488·67-s − 0.237·71-s − 0.936·73-s + 0.227·77-s + 0.225·79-s + 1.31·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.821837440\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.821837440\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.84884706738042, −13.22768349139453, −12.75673252855428, −12.14174637631999, −11.74230575937612, −11.35303275843665, −10.62873598909213, −10.21850962207738, −9.777657786317019, −9.288652336235593, −8.806736848700054, −8.204896369190435, −7.705557463213623, −7.165571105212749, −6.578572476807736, −6.174645611637261, −5.339351170116017, −5.107665861808469, −4.572548978873594, −3.903399022742443, −3.065270873997956, −2.598399107346486, −1.796762627955065, −1.507806155664351, −0.5025583542803073,
0.5025583542803073, 1.507806155664351, 1.796762627955065, 2.598399107346486, 3.065270873997956, 3.903399022742443, 4.572548978873594, 5.107665861808469, 5.339351170116017, 6.174645611637261, 6.578572476807736, 7.165571105212749, 7.705557463213623, 8.204896369190435, 8.806736848700054, 9.288652336235593, 9.777657786317019, 10.21850962207738, 10.62873598909213, 11.35303275843665, 11.74230575937612, 12.14174637631999, 12.75673252855428, 13.22768349139453, 13.84884706738042