| L(s) = 1 | + (−2.67 + 3.95i)2-s + (−1.63 − 4.93i)3-s + (−5.47 − 13.7i)4-s + (−5.58 + 12.0i)5-s + (23.8 + 6.73i)6-s + (1.76 + 6.35i)7-s + (31.6 + 6.95i)8-s + (−21.6 + 16.1i)9-s + (−32.7 − 54.4i)10-s + (−46.1 − 43.6i)11-s + (−58.7 + 49.4i)12-s + (1.00 − 9.19i)13-s + (−29.8 − 10.0i)14-s + (68.7 + 7.77i)15-s + (−26.3 + 24.9i)16-s + (−0.520 − 0.144i)17-s + ⋯ |
| L(s) = 1 | + (−0.947 + 1.39i)2-s + (−0.315 − 0.949i)3-s + (−0.683 − 1.71i)4-s + (−0.499 + 1.08i)5-s + (1.62 + 0.458i)6-s + (0.0952 + 0.342i)7-s + (1.39 + 0.307i)8-s + (−0.801 + 0.598i)9-s + (−1.03 − 1.72i)10-s + (−1.26 − 1.19i)11-s + (−1.41 + 1.19i)12-s + (0.0213 − 0.196i)13-s + (−0.569 − 0.191i)14-s + (1.18 + 0.133i)15-s + (−0.411 + 0.389i)16-s + (−0.00743 − 0.00206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.544455 + 0.0927076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.544455 + 0.0927076i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.63 + 4.93i)T \) |
| 59 | \( 1 + (334. - 305. i)T \) |
| good | 2 | \( 1 + (2.67 - 3.95i)T + (-2.96 - 7.43i)T^{2} \) |
| 5 | \( 1 + (5.58 - 12.0i)T + (-80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (-1.76 - 6.35i)T + (-293. + 176. i)T^{2} \) |
| 11 | \( 1 + (46.1 + 43.6i)T + (72.0 + 1.32e3i)T^{2} \) |
| 13 | \( 1 + (-1.00 + 9.19i)T + (-2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (0.520 + 0.144i)T + (4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (-88.2 - 67.0i)T + (1.83e3 + 6.60e3i)T^{2} \) |
| 23 | \( 1 + (4.73 + 8.93i)T + (-6.82e3 + 1.00e4i)T^{2} \) |
| 29 | \( 1 + (-210. + 142. i)T + (9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (44.9 + 59.0i)T + (-7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (52.4 + 238. i)T + (-4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (-6.81 - 3.61i)T + (3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (-265. - 280. i)T + (-4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (-540. + 249. i)T + (6.72e4 - 7.91e4i)T^{2} \) |
| 53 | \( 1 + (-53.8 + 89.4i)T + (-6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (457. + 309. i)T + (8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (61.2 - 278. i)T + (-2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (15.7 + 34.0i)T + (-2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-164. + 487. i)T + (-3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (-18.4 + 340. i)T + (-4.90e5 - 5.33e4i)T^{2} \) |
| 83 | \( 1 + (90.0 - 549. i)T + (-5.41e5 - 1.82e5i)T^{2} \) |
| 89 | \( 1 + (479. + 707. i)T + (-2.60e5 + 6.54e5i)T^{2} \) |
| 97 | \( 1 + (457. + 1.35e3i)T + (-7.26e5 + 5.52e5i)T^{2} \) |
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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
−12.20675026994882678317520957967, −11.07560316696432174731436074604, −10.27061970875107337037376114081, −8.720213925844290851219765679001, −7.81281392652144588912126276704, −7.37360695458380133407982378912, −6.13179590450700156565347093195, −5.53336336572588691884077626926, −2.86571712921514637919863555784, −0.51014190369871052315750197094,
0.891217818306391248206683338258, 2.81692322860774188455039245816, 4.28970155508186533601135244467, 5.10740613773084279938424459571, 7.49809915800030135504321872965, 8.629192148912907471423446008833, 9.367535742464030924364267860957, 10.29061016166300436969317184601, 10.92283962153278174650094952308, 12.12808800489166283257941870098
