L(s) = 1 | + 1.73i·3-s + 3.74i·5-s − 2.99·9-s − 6.48·11-s − 6.48·15-s + 3.74i·17-s − 6.92i·19-s + 6.48·23-s − 9·25-s − 5.19i·27-s + 3.46i·31-s − 11.2i·33-s − 8·37-s − 3.74i·41-s − 11.2i·45-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + 1.67i·5-s − 0.999·9-s − 1.95·11-s − 1.67·15-s + 0.907i·17-s − 1.58i·19-s + 1.35·23-s − 1.80·25-s − 0.999i·27-s + 0.622i·31-s − 1.95i·33-s − 1.31·37-s − 0.584i·41-s − 1.67i·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2781316312\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2781316312\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.74iT - 5T^{2} \) |
| 11 | \( 1 + 6.48T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 3.74iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 6.48T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 3.74iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 18.7iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.741355368428491805218475943543, −8.839248927612793042332317222188, −8.043791769808896524779598873966, −7.17025484352433245011909541038, −6.55460211356592554475328146848, −5.45339467614424216689196611988, −4.94892227586108203402590538798, −3.71782093766464687673275235877, −2.91716605522177245047374548203, −2.45038794539872617361860811970,
0.096655376359695728580090449323, 1.18272912914309086878057315245, 2.22475343223106269201023993982, 3.27696543770850333444925522531, 4.70131636481516264745009897332, 5.31621290832368697493696847915, 5.80898808755071121168295168030, 7.06380379687482077296772834254, 7.83456013884461779642792387405, 8.245555660129808595834142028905