| L(s) = 1 | + (3.62 − 2.63i)2-s + (−0.489 − 1.50i)3-s + (3.71 − 11.4i)4-s + (−16.0 − 11.6i)5-s + (−5.73 − 4.16i)6-s + (−2.16 + 6.65i)7-s + (−5.57 − 17.1i)8-s + (19.8 − 14.3i)9-s − 88.6·10-s + (34.7 + 11.1i)11-s − 19.0·12-s + (36.5 − 26.5i)13-s + (9.68 + 29.7i)14-s + (−9.68 + 29.8i)15-s + (12.5 + 9.12i)16-s + (61.3 + 44.5i)17-s + ⋯ |
| L(s) = 1 | + (1.28 − 0.930i)2-s + (−0.0941 − 0.289i)3-s + (0.464 − 1.43i)4-s + (−1.43 − 1.04i)5-s + (−0.389 − 0.283i)6-s + (−0.116 + 0.359i)7-s + (−0.246 − 0.758i)8-s + (0.733 − 0.533i)9-s − 2.80·10-s + (0.951 + 0.306i)11-s − 0.458·12-s + (0.778 − 0.565i)13-s + (0.184 + 0.568i)14-s + (−0.166 + 0.513i)15-s + (0.196 + 0.142i)16-s + (0.874 + 0.635i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.04757 - 2.10979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04757 - 2.10979i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-34.7 - 11.1i)T \) |
| good | 2 | \( 1 + (-3.62 + 2.63i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (0.489 + 1.50i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (16.0 + 11.6i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (-36.5 + 26.5i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-61.3 - 44.5i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (24.6 + 75.9i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (34.8 - 107. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (96.2 - 69.9i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-78.8 + 242. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-55.8 - 171. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 308.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-155. - 479. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-65.0 + 47.2i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (152. - 468. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (61.6 + 44.8i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 511.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (334. + 243. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-193. + 595. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (152. - 110. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-127. - 92.8i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 892.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (522. - 379. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.08168678064910035703641304694, −12.38201085948004238085652311313, −11.98452330367272565296073424444, −10.82907676946609720858195567249, −9.119309637033060992609402337382, −7.74004146273545740015030559437, −5.95179993454980435548912824617, −4.38210296500732426827310138897, −3.65744023681129593024340865636, −1.22988559482896760779508919867,
3.72048199266444541200972733788, 4.14530069155677656589826111248, 6.07414438728687744548638306771, 7.16166081562337714462253948489, 7.959559565862569804539553830225, 10.14124157405335331648675836433, 11.44118485478528899926364461734, 12.23874530756379017279489147749, 13.74700938152990945412744112953, 14.38886522225543080018174355654
