| L(s) = 1 | + (−0.513 + 3.24i)2-s + (2.81 − 1.43i)3-s + (−6.46 − 2.09i)4-s + (−4.99 + 0.0628i)5-s + (3.20 + 9.85i)6-s + (7.51 − 7.51i)7-s + (4.16 − 8.17i)8-s + (0.557 − 0.767i)9-s + (2.36 − 16.2i)10-s + (−1.41 + 1.03i)11-s + (−21.1 + 3.35i)12-s + (0.639 + 4.03i)13-s + (20.5 + 28.2i)14-s + (−13.9 + 7.33i)15-s + (2.40 + 1.75i)16-s + (−9.09 − 4.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.256 + 1.62i)2-s + (0.936 − 0.477i)3-s + (−1.61 − 0.524i)4-s + (−0.999 + 0.0125i)5-s + (0.533 + 1.64i)6-s + (1.07 − 1.07i)7-s + (0.520 − 1.02i)8-s + (0.0619 − 0.0852i)9-s + (0.236 − 1.62i)10-s + (−0.129 + 0.0937i)11-s + (−1.76 + 0.279i)12-s + (0.0491 + 0.310i)13-s + (1.46 + 2.01i)14-s + (−0.930 + 0.489i)15-s + (0.150 + 0.109i)16-s + (−0.535 − 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.703360 + 0.596926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.703360 + 0.596926i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (4.99 - 0.0628i)T \) |
| good | 2 | \( 1 + (0.513 - 3.24i)T + (-3.80 - 1.23i)T^{2} \) |
| 3 | \( 1 + (-2.81 + 1.43i)T + (5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (-7.51 + 7.51i)T - 49iT^{2} \) |
| 11 | \( 1 + (1.41 - 1.03i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-0.639 - 4.03i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (9.09 + 4.63i)T + (169. + 233. i)T^{2} \) |
| 19 | \( 1 + (23.1 - 7.51i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-24.8 - 3.93i)T + (503. + 163. i)T^{2} \) |
| 29 | \( 1 + (-0.252 - 0.0821i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-1.10 - 3.40i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-25.0 + 3.97i)T + (1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (1.53 + 1.11i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-34.0 - 34.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (20.9 + 41.0i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-2.70 + 1.37i)T + (1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (11.7 - 16.1i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-63.6 + 46.2i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (72.5 + 36.9i)T + (2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (0.716 - 2.20i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (99.1 + 15.7i)T + (5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-16.4 - 5.33i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (2.03 - 3.99i)T + (-4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (85.4 + 117. i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-63.3 - 124. i)T + (-5.53e3 + 7.61e3i)T^{2} \) |
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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
−17.38522130269229512101798685457, −16.41025838497029440799033130433, −15.00852804742731938292594712899, −14.41664206305604064446714008090, −13.27535152342757609142241735565, −11.08587923761849374642000598057, −8.724012899339784862528582168017, −7.87722302215229853812641218397, −7.04840349855560616156943584139, −4.54201882172813040774724021465,
2.68725638201871967554286120780, 4.34804264394363628347396160143, 8.363988909545162552060205518538, 9.000753369134812756912979293208, 10.79247582929932221686098320373, 11.70574626551193229133633233320, 12.89968682530286607149126277801, 14.70482003262523062044037348866, 15.44632226555050332941302186275, 17.65708687773882018584264683340
