| L(s) = 1 | − 5-s − 3i·7-s + 2·11-s + (−3 − 2i)13-s − 3·17-s − 6·23-s − 4·25-s + 6i·29-s + 3i·35-s + 3·37-s + 10i·41-s + 9i·43-s − 7i·47-s − 2·49-s − 6i·53-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13i·7-s + 0.603·11-s + (−0.832 − 0.554i)13-s − 0.727·17-s − 1.25·23-s − 0.800·25-s + 1.11i·29-s + 0.507i·35-s + 0.493·37-s + 1.56i·41-s + 1.37i·43-s − 1.02i·47-s − 0.285·49-s − 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5484268275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5484268275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + 4iT - 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−8.521627997179533620272873888961, −7.971741426722753366922585104685, −7.22895213128669574766760036898, −6.69206385819191479688383533557, −5.77697655521623084653260108503, −4.74963001396955454767547963126, −4.12096496599099372486847831525, −3.42878837016600000907123002105, −2.27692366594132568685905235280, −1.05229788046133669700911800195,
0.17471963942233941914089943808, 1.97353877672884316303680034318, 2.48037220080733794914316177378, 3.82971133827768777415195765379, 4.33348688716486543184726182852, 5.39696187218225109748207916278, 6.03935617624287427887833445924, 6.82162951728595368314451217572, 7.62046902975398087474564107542, 8.350174052216300662525422634468
