L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s + 13-s − 15-s + 16-s − 18-s + 20-s + 24-s + 25-s − 26-s − 27-s + 30-s − 32-s + 36-s − 39-s − 40-s − 2·41-s + 2·43-s + 45-s − 48-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s + 13-s − 15-s + 16-s − 18-s + 20-s + 24-s + 25-s − 26-s − 27-s + 30-s − 32-s + 36-s − 39-s − 40-s − 2·41-s + 2·43-s + 45-s − 48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6709971725\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6709971725\) |
\(L(1)\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.687436797388697297566263881654, −9.014314449018880032663446816979, −8.122087241435947886900642964907, −7.09853387789511821313589912679, −6.38182309473371025789185944231, −5.85194413889033642024200046525, −4.95344269380575781900265179921, −3.52041895484861317305858587807, −2.11733092297523868146437438628, −1.12004538889761268642888049684,
1.12004538889761268642888049684, 2.11733092297523868146437438628, 3.52041895484861317305858587807, 4.95344269380575781900265179921, 5.85194413889033642024200046525, 6.38182309473371025789185944231, 7.09853387789511821313589912679, 8.122087241435947886900642964907, 9.014314449018880032663446816979, 9.687436797388697297566263881654