| L(s) = 1 | + (−675 − 275. i)3-s − 2.11e4i·5-s − 4.02e4·7-s + (3.79e5 + 3.71e5i)9-s − 1.16e6i·11-s − 1.28e6·13-s + (−5.81e6 + 1.42e7i)15-s + 1.48e7i·17-s + 5.33e7·19-s + (2.71e7 + 1.10e7i)21-s − 1.07e8i·23-s − 2.01e8·25-s + (−1.54e8 − 3.55e8i)27-s − 1.20e8i·29-s − 6.65e7·31-s + ⋯ |
| L(s) = 1 | + (−0.925 − 0.377i)3-s − 1.35i·5-s − 0.342·7-s + (0.714 + 0.699i)9-s − 0.655i·11-s − 0.266·13-s + (−0.510 + 1.25i)15-s + 0.614i·17-s + 1.13·19-s + (0.316 + 0.129i)21-s − 0.725i·23-s − 0.825·25-s + (−0.397 − 0.917i)27-s − 0.202i·29-s − 0.0749·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.377i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.9788428898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9788428898\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (675 + 275. i)T \) |
| good | 5 | \( 1 + 2.11e4iT - 2.44e8T^{2} \) |
| 7 | \( 1 + 4.02e4T + 1.38e10T^{2} \) |
| 11 | \( 1 + 1.16e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 + 1.28e6T + 2.32e13T^{2} \) |
| 17 | \( 1 - 1.48e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 - 5.33e7T + 2.21e15T^{2} \) |
| 23 | \( 1 + 1.07e8iT - 2.19e16T^{2} \) |
| 29 | \( 1 + 1.20e8iT - 3.53e17T^{2} \) |
| 31 | \( 1 + 6.65e7T + 7.87e17T^{2} \) |
| 37 | \( 1 + 2.22e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 8.21e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 8.97e9T + 3.99e19T^{2} \) |
| 47 | \( 1 - 1.07e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 4.11e10iT - 4.91e20T^{2} \) |
| 59 | \( 1 + 4.61e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 4.06e10T + 2.65e21T^{2} \) |
| 67 | \( 1 - 1.21e11T + 8.18e21T^{2} \) |
| 71 | \( 1 + 4.48e10iT - 1.64e22T^{2} \) |
| 73 | \( 1 + 6.09e10T + 2.29e22T^{2} \) |
| 79 | \( 1 - 2.52e11T + 5.90e22T^{2} \) |
| 83 | \( 1 + 4.10e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 - 1.12e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 - 6.53e11T + 6.93e23T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.839806216506290875710143627356, −8.793833539684688066808107345202, −7.83847898606891089406428686226, −6.64722286129648163350997308779, −5.58115692082795665161865479149, −4.93844839653356353578819666989, −3.71582796017667069122052609329, −1.99646267182821997226021820883, −0.869024889047699473510334656674, −0.28559834471850539584921968889,
1.13606159254999472036538819149, 2.66041765249523946041627688295, 3.61549877800997825466109916769, 4.87031306488340426969794966160, 5.91981385792592211494785090662, 6.91781387887878662841933396416, 7.48735631304417688598171922595, 9.423869093534496612888585360335, 9.999758828021925510445561491168, 10.93610803472131105061516308227
