| 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 | \( 1 + (-12.2 - 8.88i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−14.92446920204898081478584801978, −13.05898729207809288282644258767, −11.80539371664022274477089290256, −10.71394996711341555463822321123, −9.866294396546915693825478409849, −9.478418231804577781604422826192, −8.201731346045090863671989631056, −5.64978992583445009308120468752, −4.68855134719450264845541066083, −2.38378430888767583475817264984,
0.17671981046462809713143282866, 2.12697452749309373236561182152, 5.68923233559653407565597690013, 6.89102322472560314477097221497, 7.27001994414078840858769575095, 8.623837111178989086185451059738, 9.958503484437128376293918253857, 11.20675735161051488509818096709, 12.49704371435656784008993158877, 13.61102112206651541764006430831
