| L(s) = 1 | + (158. + 274. i)2-s + (−6.86e3 + 1.18e4i)3-s + (1.51e4 − 2.63e4i)4-s + (−4.70e5 − 8.14e5i)5-s − 4.35e6·6-s + (−5.83e6 − 1.40e7i)7-s + 5.12e7·8-s + (−2.95e7 − 5.12e7i)9-s + (1.49e8 − 2.58e8i)10-s + (−3.96e7 + 6.87e7i)11-s + (2.08e8 + 3.61e8i)12-s − 8.75e8·13-s + (2.94e9 − 3.83e9i)14-s + 1.29e10·15-s + (6.13e9 + 1.06e10i)16-s + (2.41e10 − 4.19e10i)17-s + ⋯ |
| L(s) = 1 | + (0.438 + 0.758i)2-s + (−0.603 + 1.04i)3-s + (0.115 − 0.200i)4-s + (−0.538 − 0.932i)5-s − 1.05·6-s + (−0.382 − 0.924i)7-s + 1.07·8-s + (−0.229 − 0.396i)9-s + (0.471 − 0.816i)10-s + (−0.0558 + 0.0966i)11-s + (0.140 + 0.242i)12-s − 0.297·13-s + (0.533 − 0.695i)14-s + 1.29·15-s + (0.357 + 0.618i)16-s + (0.841 − 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.21508 - 0.456144i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21508 - 0.456144i\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (5.83e6 + 1.40e7i)T \) |
| good | 2 | \( 1 + (-158. - 274. i)T + (-6.55e4 + 1.13e5i)T^{2} \) |
| 3 | \( 1 + (6.86e3 - 1.18e4i)T + (-6.45e7 - 1.11e8i)T^{2} \) |
| 5 | \( 1 + (4.70e5 + 8.14e5i)T + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (3.96e7 - 6.87e7i)T + (-2.52e17 - 4.37e17i)T^{2} \) |
| 13 | \( 1 + 8.75e8T + 8.65e18T^{2} \) |
| 17 | \( 1 + (-2.41e10 + 4.19e10i)T + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (5.57e10 + 9.65e10i)T + (-2.74e21 + 4.74e21i)T^{2} \) |
| 23 | \( 1 + (-1.26e11 - 2.19e11i)T + (-7.05e22 + 1.22e23i)T^{2} \) |
| 29 | \( 1 + 4.23e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + (-2.03e12 + 3.52e12i)T + (-1.12e25 - 1.95e25i)T^{2} \) |
| 37 | \( 1 + (-5.26e11 - 9.12e11i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 - 3.11e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 3.21e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + (5.21e13 + 9.02e13i)T + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (-3.55e14 + 6.15e14i)T + (-1.02e29 - 1.77e29i)T^{2} \) |
| 59 | \( 1 + (8.77e14 - 1.52e15i)T + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (2.12e14 + 3.67e14i)T + (-1.12e30 + 1.94e30i)T^{2} \) |
| 67 | \( 1 + (2.22e15 - 3.84e15i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 - 1.95e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + (1.00e15 - 1.73e15i)T + (-2.37e31 - 4.11e31i)T^{2} \) |
| 79 | \( 1 + (-1.31e16 - 2.28e16i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 - 5.59e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + (-2.59e15 - 4.49e15i)T + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 + 7.92e16T + 5.95e33T^{2} \) |
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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
−16.87474161508514511162948122381, −16.29691718363527178275346390799, −15.18955808022977131206887628257, −13.36467050210928660171515498481, −11.23616721136512098192709953987, −9.741134431099720046613280502893, −7.30122671545342288973566675141, −5.24711480852505526120193772907, −4.28110256757916732676932881505, −0.51958843967465991595579253180,
1.83133226896544273098129167112, 3.43454905755801843249963872430, 6.19896539680432383247278884711, 7.72858557214948750529671218947, 10.72721461856274870865027771306, 12.09784532087400814329672312951, 12.76017020857304364966075096781, 14.85718515718491690040400331706, 16.85624045652916858852370569974, 18.64752583338742815029057861066
