| L(s) = 1 | + (−12.8 − 22.2i)2-s + (−179. − 311. i)3-s + (693. − 1.20e3i)4-s + 1.12e4·5-s + (−4.61e3 + 7.99e3i)6-s + (1.42e4 − 2.46e4i)7-s − 8.82e4·8-s + (2.40e4 − 4.16e4i)9-s + (−1.44e5 − 2.49e5i)10-s + (1.20e5 + 2.08e5i)11-s − 4.98e5·12-s + (−1.06e6 − 8.06e5i)13-s − 7.30e5·14-s + (−2.01e6 − 3.49e6i)15-s + (−2.87e5 − 4.97e5i)16-s + (−1.72e6 + 2.99e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.283 − 0.491i)2-s + (−0.426 − 0.739i)3-s + (0.338 − 0.586i)4-s + 1.60·5-s + (−0.242 + 0.419i)6-s + (0.319 − 0.553i)7-s − 0.952·8-s + (0.135 − 0.234i)9-s + (−0.456 − 0.790i)10-s + (0.225 + 0.391i)11-s − 0.578·12-s + (−0.798 − 0.602i)13-s − 0.363·14-s + (−0.686 − 1.18i)15-s + (−0.0684 − 0.118i)16-s + (−0.295 + 0.511i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.566960 - 1.63121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.566960 - 1.63121i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.06e6 + 8.06e5i)T \) |
| good | 2 | \( 1 + (12.8 + 22.2i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 3 | \( 1 + (179. + 311. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 - 1.12e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + (-1.42e4 + 2.46e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-1.20e5 - 2.08e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 17 | \( 1 + (1.72e6 - 2.99e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (3.90e5 - 6.76e5i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-1.37e7 - 2.37e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (1.05e7 + 1.82e7i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 - 2.43e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (3.41e8 + 5.90e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + (4.15e8 + 7.20e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (6.87e8 - 1.19e9i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 - 1.55e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.97e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (1.09e9 - 1.89e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-2.62e9 + 4.54e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-7.14e9 - 1.23e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + (-6.44e9 + 1.11e10i)T + (-1.15e20 - 2.00e20i)T^{2} \) |
| 73 | \( 1 + 1.44e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.37e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.93e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.70e10 - 2.95e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + (5.43e10 - 9.41e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−17.38525691575765424783177074362, −15.00438038147022019275964343276, −13.60588404049496395026576861516, −12.23373711232803884833903329378, −10.55203918792362445648665734862, −9.532532627789475997523065864414, −6.87373884173597115423742523753, −5.61014625413287735098367405032, −2.10623604510680653565862622888, −0.972441771902595976160844410436,
2.34928530526271900195407485812, 5.11157021645910000556117359941, 6.61648855177962990268017722128, 8.816823950303098604895197910081, 10.12791249401330836370268531684, 11.82388768546543948597786145471, 13.61406501797948276323317087306, 15.21495882385196102010361485100, 16.65926212561712216057677493161, 17.23267616674897050359718149094
