| 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
−15.52873185873094203056311615148, −14.33888953534248530644898789241, −13.15714785893593217639102511152, −11.76511713871462943837876924119, −10.26937619372938936980030887959, −9.257467084156334501149795663278, −7.966716832162022918598533594548, −6.44447107522009007461824214805, −4.95913990772314226644618251772, −1.83062455595770345762908718267,
1.65747013793858554456280040816, 4.08979385856112553124104781122, 6.35168118028768264870281905749, 7.78679884650382353883918516594, 9.346824839073704638071326775339, 10.32275243030917963633301445641, 11.38150690046539664885403876252, 12.90248009610591518899344908387, 13.92760403034399287363136046494, 15.25306408688916776459878744735
