L(s) = 1 | − 2.90·5-s − i·7-s + 1.28i·11-s + 1.86i·13-s − 3.87i·17-s + 2.04·19-s − 0.934·23-s + 3.45·25-s − 2.98·29-s + 1.85i·31-s + 2.90i·35-s − 3.85i·37-s + 8.72i·41-s + 1.19·43-s − 11.1·47-s + ⋯ |
L(s) = 1 | − 1.30·5-s − 0.377i·7-s + 0.388i·11-s + 0.518i·13-s − 0.939i·17-s + 0.468·19-s − 0.194·23-s + 0.691·25-s − 0.553·29-s + 0.333i·31-s + 0.491i·35-s − 0.634i·37-s + 1.36i·41-s + 0.181·43-s − 1.62·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.107814304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107814304\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 2.90T + 5T^{2} \) |
| 11 | \( 1 - 1.28iT - 11T^{2} \) |
| 13 | \( 1 - 1.86iT - 13T^{2} \) |
| 17 | \( 1 + 3.87iT - 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 + 0.934T + 23T^{2} \) |
| 29 | \( 1 + 2.98T + 29T^{2} \) |
| 31 | \( 1 - 1.85iT - 31T^{2} \) |
| 37 | \( 1 + 3.85iT - 37T^{2} \) |
| 41 | \( 1 - 8.72iT - 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + 10.9iT - 59T^{2} \) |
| 61 | \( 1 - 4.70iT - 61T^{2} \) |
| 67 | \( 1 + 4.41T + 67T^{2} \) |
| 71 | \( 1 + 9.72T + 71T^{2} \) |
| 73 | \( 1 - 9.40T + 73T^{2} \) |
| 79 | \( 1 - 16.4iT - 79T^{2} \) |
| 83 | \( 1 + 3.72iT - 83T^{2} \) |
| 89 | \( 1 + 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.987784933879810082990095120595, −7.32745272703215099363542540133, −6.96756473725090573371080699137, −6.01088927410428222362172073178, −4.95708247002924207533866350488, −4.46223848046850841124479861526, −3.66853771985040648596779928148, −3.00285964316588691230909348116, −1.79870857970520998743530053372, −0.57181651928980280830723857314,
0.52392351764112644112895090051, 1.80693300165186365255147772401, 2.99387432664040430268337653643, 3.63155787217996986449724919924, 4.28689960659680930828877562934, 5.21790028786421809883680490349, 5.90279048209183522253065203455, 6.69366254580728825153051554934, 7.61037630095099738338556282994, 7.949390614157186415232639655655