| L(s) = 1 | + (3.19 + 5.52i)2-s + (2.91 + 5.04i)3-s + (235. − 408. i)4-s − 1.06e3·5-s + (−18.5 + 32.2i)6-s + (3.23e3 − 5.60e3i)7-s + 6.27e3·8-s + (9.82e3 − 1.70e4i)9-s + (−3.39e3 − 5.88e3i)10-s + (−9.31e3 − 1.61e4i)11-s + 2.74e3·12-s + (7.63e4 − 6.90e4i)13-s + 4.13e4·14-s + (−3.09e3 − 5.36e3i)15-s + (−1.00e5 − 1.74e5i)16-s + (−2.60e5 + 4.52e5i)17-s + ⋯ |
| L(s) = 1 | + (0.141 + 0.244i)2-s + (0.0207 + 0.0359i)3-s + (0.460 − 0.797i)4-s − 0.761·5-s + (−0.00585 + 0.0101i)6-s + (0.509 − 0.882i)7-s + 0.541·8-s + (0.499 − 0.864i)9-s + (−0.107 − 0.185i)10-s + (−0.191 − 0.332i)11-s + 0.0382·12-s + (0.741 − 0.670i)13-s + 0.287·14-s + (−0.0158 − 0.0273i)15-s + (−0.383 − 0.664i)16-s + (−0.757 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.50181 - 0.911191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50181 - 0.911191i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-7.63e4 + 6.90e4i)T \) |
| good | 2 | \( 1 + (-3.19 - 5.52i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (-2.91 - 5.04i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + 1.06e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + (-3.23e3 + 5.60e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (9.31e3 + 1.61e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 17 | \( 1 + (2.60e5 - 4.52e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.36e5 - 2.37e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-4.42e5 - 7.67e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (1.89e6 + 3.27e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 - 9.87e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (-5.10e6 - 8.84e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-6.37e6 - 1.10e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (2.01e6 - 3.49e6i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 - 1.84e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.32e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-8.32e6 + 1.44e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-4.50e7 + 7.79e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.30e7 + 2.25e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.40e8 - 2.42e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 - 1.22e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.73e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (6.13e7 + 1.06e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (1.54e8 - 2.67e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−17.38739583197867975202272254005, −15.71415516789454939450454471482, −14.99465133376969635252997285163, −13.38237422398847672088447366298, −11.43615967380529443095532286541, −10.25782641757604214318167310958, −7.961279999318975844122005836111, −6.30130408858647237033894471436, −4.09573368744352287510834433970, −1.02391604874757114450532487375,
2.34329966717100102972544402589, 4.49428699417159863637617750974, 7.20719384804755308420776295939, 8.590412995038826874338153360985, 11.07210209192433145988525756964, 12.02284421561984009502331059125, 13.47652568288825034729775560652, 15.48066865405650025967289282079, 16.30418223244241988854972383950, 18.04288638886111515186374722593
