| L(s) = 1 | + (0.711 + 4.49i)3-s + (−4.53 + 2.09i)5-s + (−3.58 + 3.58i)7-s + (−11.1 + 3.61i)9-s + (−3.53 + 10.8i)11-s + (10.0 − 19.7i)13-s + (−12.6 − 18.9i)15-s + (−3.10 + 19.6i)17-s + (−15.8 − 21.8i)19-s + (−18.6 − 13.5i)21-s + (0.476 − 0.242i)23-s + (16.2 − 19.0i)25-s + (−5.57 − 10.9i)27-s + (−3.67 + 5.05i)29-s + (−5.42 + 3.94i)31-s + ⋯ |
| L(s) = 1 | + (0.237 + 1.49i)3-s + (−0.907 + 0.419i)5-s + (−0.512 + 0.512i)7-s + (−1.23 + 0.401i)9-s + (−0.321 + 0.989i)11-s + (0.773 − 1.51i)13-s + (−0.843 − 1.26i)15-s + (−0.182 + 1.15i)17-s + (−0.833 − 1.14i)19-s + (−0.889 − 0.645i)21-s + (0.0207 − 0.0105i)23-s + (0.648 − 0.761i)25-s + (−0.206 − 0.405i)27-s + (−0.126 + 0.174i)29-s + (−0.174 + 0.127i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.243129 - 0.572754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.243129 - 0.572754i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (4.53 - 2.09i)T \) |
| good | 3 | \( 1 + (-0.711 - 4.49i)T + (-8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (3.58 - 3.58i)T - 49iT^{2} \) |
| 11 | \( 1 + (3.53 - 10.8i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-10.0 + 19.7i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (3.10 - 19.6i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (15.8 + 21.8i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-0.476 + 0.242i)T + (310. - 427. i)T^{2} \) |
| 29 | \( 1 + (3.67 - 5.05i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (5.42 - 3.94i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (5.24 + 2.67i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-7.33 - 22.5i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (44.7 + 44.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-27.2 + 4.31i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (13.6 + 86.0i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (20.7 - 6.73i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (21.3 - 65.6i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (14.4 - 91.1i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (12.7 + 9.24i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (60.4 - 30.7i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (71.8 - 98.9i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (60.8 + 9.64i)T + (6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-75.7 - 24.6i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (134. - 21.2i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−11.28774757557098036221697212126, −10.53956294635572602463962816088, −10.06880415569982127219953991096, −8.851707903493192033209956986866, −8.265989697983820874725102971857, −6.97483660895081210592809480953, −5.69291826319346377161355532002, −4.57130155172876821510306024308, −3.68372019983188998800712848603, −2.76306183709229177325177147992,
0.26276015431144570281206064547, 1.58732175179921439129485179417, 3.20171681194596427839048464204, 4.34555279241909787952732131007, 6.03680938974098317705577342589, 6.82475396435177869903125480248, 7.65912951206176362916155609174, 8.436533399255920194426012408714, 9.236682599787694836254981063480, 10.79176096517744025230594773480
