| L(s) = 1 | + (3.43 − 7.13i)2-s + (−5.05 − 0.378i)3-s + (−29.0 − 36.4i)4-s + (−5.90 + 19.1i)5-s + (−20.0 + 34.7i)6-s + (59.2 − 34.1i)7-s + (−236. + 53.9i)8-s + (−54.7 − 8.24i)9-s + (116. + 107. i)10-s + (78.0 − 97.8i)11-s + (133. + 195. i)12-s + (112. − 104. i)13-s + (−40.4 − 539. i)14-s + (37.0 − 94.5i)15-s + (−260. + 1.14e3i)16-s + (−98.9 + 30.5i)17-s + ⋯ |
| L(s) = 1 | + (0.858 − 1.78i)2-s + (−0.561 − 0.0420i)3-s + (−1.81 − 2.27i)4-s + (−0.236 + 0.766i)5-s + (−0.556 + 0.964i)6-s + (1.20 − 0.697i)7-s + (−3.69 + 0.843i)8-s + (−0.675 − 0.101i)9-s + (1.16 + 1.07i)10-s + (0.645 − 0.808i)11-s + (0.923 + 1.35i)12-s + (0.666 − 0.618i)13-s + (−0.206 − 2.75i)14-s + (0.164 − 0.420i)15-s + (−1.01 + 4.46i)16-s + (−0.342 + 0.105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.169882 + 1.55369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.169882 + 1.55369i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-89.9 - 1.84e3i)T \) |
| good | 2 | \( 1 + (-3.43 + 7.13i)T + (-9.97 - 12.5i)T^{2} \) |
| 3 | \( 1 + (5.05 + 0.378i)T + (80.0 + 12.0i)T^{2} \) |
| 5 | \( 1 + (5.90 - 19.1i)T + (-516. - 352. i)T^{2} \) |
| 7 | \( 1 + (-59.2 + 34.1i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-78.0 + 97.8i)T + (-3.25e3 - 1.42e4i)T^{2} \) |
| 13 | \( 1 + (-112. + 104. i)T + (2.13e3 - 2.84e4i)T^{2} \) |
| 17 | \( 1 + (98.9 - 30.5i)T + (6.90e4 - 4.70e4i)T^{2} \) |
| 19 | \( 1 + (55.9 + 371. i)T + (-1.24e5 + 3.84e4i)T^{2} \) |
| 23 | \( 1 + (-37.7 - 96.1i)T + (-2.05e5 + 1.90e5i)T^{2} \) |
| 29 | \( 1 + (-428. + 32.0i)T + (6.99e5 - 1.05e5i)T^{2} \) |
| 31 | \( 1 + (-772. + 526. i)T + (3.37e5 - 8.59e5i)T^{2} \) |
| 37 | \( 1 + (-320. - 185. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (647. + 311. i)T + (1.76e6 + 2.20e6i)T^{2} \) |
| 47 | \( 1 + (-544. - 682. i)T + (-1.08e6 + 4.75e6i)T^{2} \) |
| 53 | \( 1 + (84.7 + 78.6i)T + (5.89e5 + 7.86e6i)T^{2} \) |
| 59 | \( 1 + (1.27e3 - 5.57e3i)T + (-1.09e7 - 5.25e6i)T^{2} \) |
| 61 | \( 1 + (-1.14e3 + 1.67e3i)T + (-5.05e6 - 1.28e7i)T^{2} \) |
| 67 | \( 1 + (6.26e3 - 944. i)T + (1.92e7 - 5.93e6i)T^{2} \) |
| 71 | \( 1 + (-5.25e3 - 2.06e3i)T + (1.86e7 + 1.72e7i)T^{2} \) |
| 73 | \( 1 + (4.23e3 + 4.55e3i)T + (-2.12e6 + 2.83e7i)T^{2} \) |
| 79 | \( 1 + (2.39e3 + 4.14e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-429. + 5.72e3i)T + (-4.69e7 - 7.07e6i)T^{2} \) |
| 89 | \( 1 + (-199. - 14.9i)T + (6.20e7 + 9.35e6i)T^{2} \) |
| 97 | \( 1 + (-721. + 905. i)T + (-1.96e7 - 8.63e7i)T^{2} \) |
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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
−14.19441986819003082024073694290, −13.35598990568938679221485302118, −11.71532549339878968727661918926, −11.17391353314474147151408033198, −10.58161787901650523728705468060, −8.712893877724436248603422909582, −6.07898772467978511934650571666, −4.58853367451182978079925457187, −3.07752457995871993060684198169, −0.916314707710024516521181517995,
4.38070937172169979664658622071, 5.28782346433406060732458331594, 6.52514047828025727064718514997, 8.198686731117152505409259267439, 8.850754099429917844710346397310, 11.76709179099821818331284909756, 12.39501870329524772779933378339, 13.96964435367536861055459575302, 14.71785094833842956455302543611, 15.80739684564112039944411452617
