| L(s) = 1 | + (3.33 − 0.894i)2-s + (6.96 + 1.86i)3-s + (−3.50 + 2.02i)4-s + (21.4 − 12.8i)5-s + 24.9·6-s + (48.0 + 9.66i)7-s + (−49.0 + 49.0i)8-s + (−25.1 − 14.5i)9-s + (60.1 − 62.0i)10-s + (−39.5 − 68.4i)11-s + (−28.1 + 7.55i)12-s + (−37.2 + 37.2i)13-s + (169. − 10.7i)14-s + (173. − 49.2i)15-s + (−87.4 + 151. i)16-s + (−55.3 + 206. i)17-s + ⋯ |
| L(s) = 1 | + (0.834 − 0.223i)2-s + (0.773 + 0.207i)3-s + (−0.219 + 0.126i)4-s + (0.858 − 0.513i)5-s + 0.692·6-s + (0.980 + 0.197i)7-s + (−0.765 + 0.765i)8-s + (−0.310 − 0.179i)9-s + (0.601 − 0.620i)10-s + (−0.326 − 0.565i)11-s + (−0.195 + 0.0524i)12-s + (−0.220 + 0.220i)13-s + (0.862 − 0.0546i)14-s + (0.770 − 0.219i)15-s + (−0.341 + 0.591i)16-s + (−0.191 + 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0897i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 + 0.0897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.54986 - 0.114667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.54986 - 0.114667i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-21.4 + 12.8i)T \) |
| 7 | \( 1 + (-48.0 - 9.66i)T \) |
| good | 2 | \( 1 + (-3.33 + 0.894i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (-6.96 - 1.86i)T + (70.1 + 40.5i)T^{2} \) |
| 11 | \( 1 + (39.5 + 68.4i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (37.2 - 37.2i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (55.3 - 206. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (615. + 355. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-110. - 412. i)T + (-2.42e5 + 1.39e5i)T^{2} \) |
| 29 | \( 1 - 237. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (-567. - 983. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-1.11e3 + 299. i)T + (1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 - 1.40e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-2.19e3 + 2.19e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-793. + 212. i)T + (4.22e6 - 2.43e6i)T^{2} \) |
| 53 | \( 1 + (1.85e3 + 498. i)T + (6.83e6 + 3.94e6i)T^{2} \) |
| 59 | \( 1 + (3.06e3 - 1.76e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (433. - 751. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.77e3 - 6.62e3i)T + (-1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 + 486.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-1.82e3 - 488. i)T + (2.45e7 + 1.41e7i)T^{2} \) |
| 79 | \( 1 + (-5.53e3 - 3.19e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (325. - 325. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (3.04e3 + 1.75e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (4.80e3 + 4.80e3i)T + 8.85e7iT^{2} \) |
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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
−15.23306025302097895670127705790, −14.32509650270791905212749533209, −13.55036589871350535798511505114, −12.47635089738013137296312362736, −10.98657407831394151016642831263, −9.040055756371419425794528985449, −8.423677971731035357649185556275, −5.76784665720349032947798481756, −4.39704830129841880601650834242, −2.48009615208854528470718805307,
2.42903936450073476676803463890, 4.59075755318928619793868452256, 6.11519867800199586511783716071, 7.895653237147295993501007337167, 9.397696558230828667175351358040, 10.79806760354277722817768160214, 12.68563760463494353801399607836, 13.71834271907607412577540711785, 14.51310332511298254149025252483, 15.03721998845949461144737070703
