L(s) = 1 | + 1.73·3-s − 3.46i·5-s − 2i·7-s + 2.99·9-s − 3.46·11-s − 5.99i·15-s − 3.46i·21-s − 6.99·25-s + 5.19·27-s − 10.3i·29-s + 10i·31-s − 5.99·33-s − 6.92·35-s − 10.3i·45-s + 3·49-s + ⋯ |
L(s) = 1 | + 1.00·3-s − 1.54i·5-s − 0.755i·7-s + 0.999·9-s − 1.04·11-s − 1.54i·15-s − 0.755i·21-s − 1.39·25-s + 1.00·27-s − 1.92i·29-s + 1.79i·31-s − 1.04·33-s − 1.17·35-s − 1.54i·45-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40258 - 1.40258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40258 - 1.40258i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.930397457547826135757495616788, −9.194464113499630952432481745675, −8.277080756654716251741115385638, −7.909711051869976335881813279343, −6.83324376074904436367170150595, −5.36615790469085580561597214526, −4.58051703211354682183803819234, −3.69222820900469522900583935906, −2.27893758085900756219810726486, −0.912602202984102835619506873527,
2.19664772209953578996182997816, 2.83838686951482260099094827100, 3.74800611960441040979023191349, 5.19522761895446153394699703063, 6.32327614962902531203610541779, 7.23716426122048259564048818696, 7.87088267455179116597779560720, 8.822242900305511294082045724310, 9.715401122909150904043729428708, 10.48971901354522302832364767553