| L(s) = 1 | + (−1.53 + 1.28i)2-s + (5.19 + 4.35i)3-s + (0.694 − 3.93i)4-s + (−18.6 + 6.79i)5-s − 13.5·6-s + (−8.13 + 2.96i)7-s + (4.00 + 6.92i)8-s + (3.28 + 18.6i)9-s + (19.8 − 34.3i)10-s + (−1.63 − 2.82i)11-s + (20.7 − 17.4i)12-s + (−3.84 + 21.8i)13-s + (8.66 − 15.0i)14-s + (−126. − 46.0i)15-s + (−15.0 − 5.47i)16-s + (13.0 + 73.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.999 + 0.838i)3-s + (0.0868 − 0.492i)4-s + (−1.66 + 0.607i)5-s − 0.922·6-s + (−0.439 + 0.159i)7-s + (0.176 + 0.306i)8-s + (0.121 + 0.690i)9-s + (0.628 − 1.08i)10-s + (−0.0447 − 0.0774i)11-s + (0.499 − 0.419i)12-s + (−0.0820 + 0.465i)13-s + (0.165 − 0.286i)14-s + (−2.17 − 0.792i)15-s + (−0.234 − 0.0855i)16-s + (0.185 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0730810 + 0.805683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0730810 + 0.805683i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.53 - 1.28i)T \) |
| 37 | \( 1 + (64.2 - 215. i)T \) |
| good | 3 | \( 1 + (-5.19 - 4.35i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (18.6 - 6.79i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (8.13 - 2.96i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (1.63 + 2.82i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (3.84 - 21.8i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-13.0 - 73.7i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 19 | \( 1 + (51.6 + 43.3i)T + (1.19e3 + 6.75e3i)T^{2} \) |
| 23 | \( 1 + (81.8 - 141. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-110. - 191. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 255.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-52.1 + 295. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 - 129.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (210. - 364. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-153. - 55.7i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-319. - 116. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-107. + 609. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-637. + 231. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (619. + 519. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + 615.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-73.1 + 26.6i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-114. - 647. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (1.36e3 + 495. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (550. - 953. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.95414796546085979888131104464, −14.03266529162516357857578944988, −12.23616777308854490363863354582, −11.00405624288942237384885543924, −9.935150382308938093568105877203, −8.687483424728026139299640980758, −7.942025584903221933491333350738, −6.63411589861914632790901341713, −4.30402799729845847652893728662, −3.17074110555529788258754703731,
0.55260931392001279066316856317, 2.77571605532819863354831386088, 4.22837096883283631068116550143, 7.00450098415705807393258574302, 8.089605511479195813617970759311, 8.485565674113907497441933856150, 10.05808100507839839695599819002, 11.63532927522155957087871319299, 12.40996817616584433486411004061, 13.26912714316252338362861677590
