| L(s) = 1 | + (−1.68 − 1.08i)2-s + (0.0510 + 0.354i)3-s + (1.66 + 3.63i)4-s + (8.79 + 2.58i)5-s + (0.297 − 0.652i)6-s + (10.6 − 12.3i)7-s + (1.13 − 7.91i)8-s + (25.7 − 7.57i)9-s + (−12.0 − 13.8i)10-s + (11.8 − 7.61i)11-s + (−1.20 + 0.775i)12-s + (10.2 + 11.8i)13-s + (−31.3 + 9.19i)14-s + (−0.467 + 3.25i)15-s + (−10.4 + 12.0i)16-s + (−28.7 + 62.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (0.00981 + 0.0682i)3-s + (0.207 + 0.454i)4-s + (0.786 + 0.230i)5-s + (0.0202 − 0.0443i)6-s + (0.577 − 0.666i)7-s + (0.0503 − 0.349i)8-s + (0.954 − 0.280i)9-s + (−0.379 − 0.437i)10-s + (0.324 − 0.208i)11-s + (−0.0290 + 0.0186i)12-s + (0.218 + 0.252i)13-s + (−0.598 + 0.175i)14-s + (−0.00804 + 0.0559i)15-s + (−0.163 + 0.188i)16-s + (−0.410 + 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.20080 - 0.262373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20080 - 0.262373i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.68 + 1.08i)T \) |
| 23 | \( 1 + (86.3 - 68.6i)T \) |
| good | 3 | \( 1 + (-0.0510 - 0.354i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (-8.79 - 2.58i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-10.6 + 12.3i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-11.8 + 7.61i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-10.2 - 11.8i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (28.7 - 62.9i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (10.3 + 22.5i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (59.2 - 129. i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-40.0 + 278. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-114. + 33.6i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (26.6 + 7.83i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (6.67 + 46.4i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + 520.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (279. - 322. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (37.1 + 42.8i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (34.5 - 240. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (720. + 462. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-156. - 100. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-370. - 810. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-117. - 135. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-661. + 194. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-102. - 711. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (1.19e3 + 351. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−15.25306408688916776459878744735, −13.92760403034399287363136046494, −12.90248009610591518899344908387, −11.38150690046539664885403876252, −10.32275243030917963633301445641, −9.346824839073704638071326775339, −7.78679884650382353883918516594, −6.35168118028768264870281905749, −4.08979385856112553124104781122, −1.65747013793858554456280040816,
1.83062455595770345762908718267, 4.95913990772314226644618251772, 6.44447107522009007461824214805, 7.966716832162022918598533594548, 9.257467084156334501149795663278, 10.26937619372938936980030887959, 11.76511713871462943837876924119, 13.15714785893593217639102511152, 14.33888953534248530644898789241, 15.52873185873094203056311615148
