| L(s) = 1 | + (2.13 + 2.54i)2-s + (−0.592 + 1.62i)3-s + (−1.22 + 6.92i)4-s + (−0.870 − 4.93i)5-s + (−5.40 + 1.96i)6-s + (2.46 − 4.27i)7-s + (−8.70 + 5.02i)8-s + (−2.29 − 1.92i)9-s + (10.7 − 12.7i)10-s + (7.89 + 13.6i)11-s + (−10.5 − 6.08i)12-s + (−4.52 − 12.4i)13-s + (16.1 − 2.84i)14-s + (8.55 + 1.50i)15-s + (−4.96 − 1.80i)16-s + (−18.3 + 15.3i)17-s + ⋯ |
| L(s) = 1 | + (1.06 + 1.27i)2-s + (−0.197 + 0.542i)3-s + (−0.305 + 1.73i)4-s + (−0.174 − 0.987i)5-s + (−0.900 + 0.327i)6-s + (0.352 − 0.610i)7-s + (−1.08 + 0.628i)8-s + (−0.255 − 0.214i)9-s + (1.07 − 1.27i)10-s + (0.717 + 1.24i)11-s + (−0.878 − 0.507i)12-s + (−0.348 − 0.957i)13-s + (1.15 − 0.203i)14-s + (0.570 + 0.100i)15-s + (−0.310 − 0.112i)16-s + (−1.07 + 0.904i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.16563 + 1.32734i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16563 + 1.32734i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.592 - 1.62i)T \) |
| 19 | \( 1 + (11.5 + 15.0i)T \) |
| good | 2 | \( 1 + (-2.13 - 2.54i)T + (-0.694 + 3.93i)T^{2} \) |
| 5 | \( 1 + (0.870 + 4.93i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-2.46 + 4.27i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.89 - 13.6i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.52 + 12.4i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (18.3 - 15.3i)T + (50.1 - 284. i)T^{2} \) |
| 23 | \( 1 + (-4.49 + 25.5i)T + (-497. - 180. i)T^{2} \) |
| 29 | \( 1 + (9.60 - 11.4i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (32.4 + 18.7i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 22.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (20.3 - 55.8i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-9.77 - 55.4i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-58.7 - 49.2i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + (28.6 + 5.05i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (21.0 + 25.0i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-3.90 + 22.1i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-6.43 + 7.67i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-7.22 + 1.27i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-24.7 - 9.02i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-31.1 + 85.7i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-68.8 + 119. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-8.75 - 24.0i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-78.1 - 93.0i)T + (-1.63e3 + 9.26e3i)T^{2} \) |
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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
−15.10186725969666342477990614510, −14.60937804907273525299565490994, −13.01194615433677781019417601651, −12.54223997840961131188700811772, −10.78176535367868153495077047430, −9.057873664917771931572281030734, −7.75409324566139806140939003456, −6.42581090913140220542129393539, −4.81158773766257856888838218597, −4.27705891995767132590495005095,
2.15048373956594913021467897034, 3.69134121415097667686204910641, 5.52246136668437744535174058420, 6.92048373975994271379792347372, 8.984216033310957617988706886862, 10.74822802956293397331349855593, 11.48777482524095166983189040372, 12.12207358795479205864726991244, 13.62383952514405521005921112934, 14.16122542917812217650208970843
