| L(s) = 1 | + (−46.2 − 46.2i)2-s + (−297. − 717. i)3-s + 2.22e3i·4-s + (146. − 60.6i)5-s + (−1.94e4 + 4.68e4i)6-s + (6.84e4 + 2.83e4i)7-s + (8.10e3 − 8.10e3i)8-s + (−3.00e5 + 3.00e5i)9-s + (−9.56e3 − 3.96e3i)10-s + (6.46e4 − 1.56e5i)11-s + (1.59e6 − 6.60e5i)12-s + 9.36e5i·13-s + (−1.85e6 − 4.47e6i)14-s + (−8.69e4 − 8.69e4i)15-s + 3.80e6·16-s + (−2.14e6 + 5.44e6i)17-s + ⋯ |
| L(s) = 1 | + (−1.02 − 1.02i)2-s + (−0.705 − 1.70i)3-s + 1.08i·4-s + (0.0209 − 0.00867i)5-s + (−1.01 + 2.46i)6-s + (1.53 + 0.637i)7-s + (0.0874 − 0.0874i)8-s + (−1.69 + 1.69i)9-s + (−0.0302 − 0.0125i)10-s + (0.121 − 0.292i)11-s + (1.84 − 0.766i)12-s + 0.699i·13-s + (−0.920 − 2.22i)14-s + (−0.0295 − 0.0295i)15-s + 0.907·16-s + (−0.366 + 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.498778 - 0.114569i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.498778 - 0.114569i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (2.14e6 - 5.44e6i)T \) |
| good | 2 | \( 1 + (46.2 + 46.2i)T + 2.04e3iT^{2} \) |
| 3 | \( 1 + (297. + 717. i)T + (-1.25e5 + 1.25e5i)T^{2} \) |
| 5 | \( 1 + (-146. + 60.6i)T + (3.45e7 - 3.45e7i)T^{2} \) |
| 7 | \( 1 + (-6.84e4 - 2.83e4i)T + (1.39e9 + 1.39e9i)T^{2} \) |
| 11 | \( 1 + (-6.46e4 + 1.56e5i)T + (-2.01e11 - 2.01e11i)T^{2} \) |
| 13 | \( 1 - 9.36e5iT - 1.79e12T^{2} \) |
| 19 | \( 1 + (1.35e5 + 1.35e5i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + (2.86e6 - 6.91e6i)T + (-6.73e14 - 6.73e14i)T^{2} \) |
| 29 | \( 1 + (1.97e8 - 8.17e7i)T + (8.62e15 - 8.62e15i)T^{2} \) |
| 31 | \( 1 + (-7.82e6 - 1.88e7i)T + (-1.79e16 + 1.79e16i)T^{2} \) |
| 37 | \( 1 + (-1.21e8 - 2.92e8i)T + (-1.25e17 + 1.25e17i)T^{2} \) |
| 41 | \( 1 + (-3.03e8 - 1.25e8i)T + (3.89e17 + 3.89e17i)T^{2} \) |
| 43 | \( 1 + (-9.11e8 + 9.11e8i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 - 2.02e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + (3.33e9 + 3.33e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (-3.03e9 + 3.03e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (-5.36e9 - 2.22e9i)T + (3.07e19 + 3.07e19i)T^{2} \) |
| 67 | \( 1 + 1.50e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-1.87e9 - 4.53e9i)T + (-1.63e20 + 1.63e20i)T^{2} \) |
| 73 | \( 1 + (1.68e10 - 6.96e9i)T + (2.21e20 - 2.21e20i)T^{2} \) |
| 79 | \( 1 + (1.14e10 - 2.76e10i)T + (-5.28e20 - 5.28e20i)T^{2} \) |
| 83 | \( 1 + (2.06e10 + 2.06e10i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 + 2.70e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + (1.89e10 - 7.85e9i)T + (5.05e21 - 5.05e21i)T^{2} \) |
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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
−17.13226753044570780793734509924, −14.45392933922748570741937009020, −12.80514529885734173440616875906, −11.56112321991980654093779209788, −11.15876665308594387167367279144, −8.749309261670922293618358964907, −7.66301114194476025069524725057, −5.71116625972259642702690206757, −2.01899661284663681834277995134, −1.38204157639810523224466272018,
0.35783202620189160274765387378, 4.29570893798262137244545476757, 5.66137124169884081790155360485, 7.71353495878085888845094019803, 9.201091057301849955497713650975, 10.37302300415962183182527200578, 11.47185220854825945819845560319, 14.51326949838680644291521902181, 15.38145154747809158858535395277, 16.43031584547765841523135380137
