| L(s) = 1 | + (−3.23 + 2.35i)2-s + (−10.5 − 7.64i)3-s + (4.94 − 15.2i)4-s − 86.6·5-s + 52.0·6-s + (−58.6 + 180. i)7-s + (19.7 + 60.8i)8-s + (−22.8 − 70.2i)9-s + (280. − 203. i)10-s + (211. − 649. i)11-s + (−168. + 122. i)12-s + (363. + 263. i)13-s + (−234. − 721. i)14-s + (911. + 662. i)15-s + (−207. − 150. i)16-s + (562. + 1.73e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.674 − 0.490i)3-s + (0.154 − 0.475i)4-s − 1.55·5-s + 0.589·6-s + (−0.452 + 1.39i)7-s + (0.109 + 0.336i)8-s + (−0.0939 − 0.289i)9-s + (0.886 − 0.644i)10-s + (0.526 − 1.61i)11-s + (−0.337 + 0.245i)12-s + (0.596 + 0.433i)13-s + (−0.319 − 0.984i)14-s + (1.04 + 0.760i)15-s + (−0.202 − 0.146i)16-s + (0.471 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.600810 + 0.0756018i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.600810 + 0.0756018i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 31 | \( 1 + (420. - 5.33e3i)T \) |
| good | 3 | \( 1 + (10.5 + 7.64i)T + (75.0 + 231. i)T^{2} \) |
| 5 | \( 1 + 86.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + (58.6 - 180. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 11 | \( 1 + (-211. + 649. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-363. - 263. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-562. - 1.73e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-841. + 611. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (515. + 1.58e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.75e3 + 3.45e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 37 | \( 1 + 1.02e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-9.18e3 + 6.67e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + (-484. + 351. i)T + (4.54e7 - 1.39e8i)T^{2} \) |
| 47 | \( 1 + (-1.11e4 - 8.13e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (4.76e3 + 1.46e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-5.10e3 - 3.70e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + 1.58e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.54e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-1.73e4 - 5.33e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-1.38e4 + 4.25e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.35e4 - 7.25e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-9.52e3 + 6.91e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-3.63e4 + 1.11e5i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (2.64e4 - 8.14e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−14.31978410572132552711773956295, −12.44952370508908112874041238592, −11.80503742006063293086115845298, −10.93663697587822953654856286304, −8.878896984731317899533440752594, −8.294849984013591405524472510680, −6.61718928307255184633647173943, −5.78946394131033812303253949567, −3.50157644666861500976442570535, −0.72531151548418303699267073744,
0.67645138436783390492998782194, 3.60204107670837653196485803913, 4.65455694276491695088691497245, 7.11568248887444510143971592300, 7.78873572445977414754326773027, 9.660061309218194876393701564689, 10.58158534979349821571621806155, 11.55001554392403548556136328993, 12.35624995231544139728423088829, 13.90380355848715732950720554378
