| L(s) = 1 | + (−3.08 + 2.24i)2-s + (−2.65 − 8.17i)3-s + (2.03 − 6.25i)4-s + (9.24 + 6.71i)5-s + (26.5 + 19.2i)6-s + (2.16 − 6.65i)7-s + (−1.68 − 5.18i)8-s + (−37.9 + 27.5i)9-s − 43.6·10-s + (−24.0 − 27.4i)11-s − 56.4·12-s + (−59.7 + 43.3i)13-s + (8.25 + 25.4i)14-s + (30.3 − 93.3i)15-s + (59.3 + 43.1i)16-s + (−88.5 − 64.3i)17-s + ⋯ |
| L(s) = 1 | + (−1.09 + 0.793i)2-s + (−0.511 − 1.57i)3-s + (0.253 − 0.781i)4-s + (0.826 + 0.600i)5-s + (1.80 + 1.31i)6-s + (0.116 − 0.359i)7-s + (−0.0744 − 0.229i)8-s + (−1.40 + 1.01i)9-s − 1.37·10-s + (−0.658 − 0.752i)11-s − 1.35·12-s + (−1.27 + 0.925i)13-s + (0.157 + 0.485i)14-s + (0.522 − 1.60i)15-s + (0.927 + 0.673i)16-s + (−1.26 − 0.918i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0378656 - 0.196746i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0378656 - 0.196746i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.16 + 6.65i)T \) |
| 11 | \( 1 + (24.0 + 27.4i)T \) |
| good | 2 | \( 1 + (3.08 - 2.24i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (2.65 + 8.17i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-9.24 - 6.71i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (59.7 - 43.3i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (88.5 + 64.3i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-12.8 - 39.6i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 74.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + (74.7 - 230. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-2.19 + 1.59i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-63.1 + 194. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (106. + 326. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 6.07T + 7.95e4T^{2} \) |
| 47 | \( 1 + (29.2 + 90.0i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-364. + 264. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-12.2 + 37.8i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-163. - 118. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 609.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (48.7 + 35.4i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (213. - 657. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-374. + 272. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (1.03e3 + 753. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-12.2 + 8.88i)T + (2.82e5 - 8.68e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.61102112206651541764006430831, −12.49704371435656784008993158877, −11.20675735161051488509818096709, −9.958503484437128376293918253857, −8.623837111178989086185451059738, −7.27001994414078840858769575095, −6.89102322472560314477097221497, −5.68923233559653407565597690013, −2.12697452749309373236561182152, −0.17671981046462809713143282866,
2.38378430888767583475817264984, 4.68855134719450264845541066083, 5.64978992583445009308120468752, 8.201731346045090863671989631056, 9.478418231804577781604422826192, 9.866294396546915693825478409849, 10.71394996711341555463822321123, 11.80539371664022274477089290256, 13.05898729207809288282644258767, 14.92446920204898081478584801978
