| L(s) = 1 | + (1.84 + 3.20i)3-s + (−2.68 + 2.68i)5-s + (3.21 − 0.861i)7-s + (−2.33 + 4.04i)9-s + (3.45 − 12.9i)11-s + (10.7 + 7.37i)13-s + (−13.5 − 3.63i)15-s + (−18.0 − 10.4i)17-s + (−8.17 − 30.5i)19-s + (8.70 + 8.70i)21-s + (−20.2 + 11.6i)23-s + 10.5i·25-s + 16.0·27-s + (16.7 + 28.9i)29-s + (15.6 − 15.6i)31-s + ⋯ |
| L(s) = 1 | + (0.616 + 1.06i)3-s + (−0.537 + 0.537i)5-s + (0.459 − 0.123i)7-s + (−0.259 + 0.449i)9-s + (0.314 − 1.17i)11-s + (0.823 + 0.567i)13-s + (−0.904 − 0.242i)15-s + (−1.06 − 0.612i)17-s + (−0.430 − 1.60i)19-s + (0.414 + 0.414i)21-s + (−0.879 + 0.507i)23-s + 0.423i·25-s + 0.592·27-s + (0.576 + 0.999i)29-s + (0.505 − 0.505i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15769 + 0.589430i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15769 + 0.589430i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (-10.7 - 7.37i)T \) |
| good | 3 | \( 1 + (-1.84 - 3.20i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (2.68 - 2.68i)T - 25iT^{2} \) |
| 7 | \( 1 + (-3.21 + 0.861i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (-3.45 + 12.9i)T + (-104. - 60.5i)T^{2} \) |
| 17 | \( 1 + (18.0 + 10.4i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (8.17 + 30.5i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (20.2 - 11.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-16.7 - 28.9i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-15.6 + 15.6i)T - 961iT^{2} \) |
| 37 | \( 1 + (4.36 - 16.3i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (58.8 + 15.7i)T + (1.45e3 + 840.5i)T^{2} \) |
| 43 | \( 1 + (-43.5 - 25.1i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-1.03 - 1.03i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + 63.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (26.8 - 7.18i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-46.2 + 80.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (40.1 + 10.7i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-15.6 - 58.4i)T + (-4.36e3 + 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-36.8 - 36.8i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 106.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-96.6 + 96.6i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (6.66 - 24.8i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-4.20 - 15.6i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−15.54955156471610606425839577969, −14.35857579870410435810828096192, −13.52988795900335620331907531181, −11.43812127006421449225440676030, −10.89243136911555121644248213682, −9.289191736964084459091357809051, −8.419708569529574246690572785665, −6.67165393483019533868739001122, −4.54240951887410995657666408738, −3.26698574055719600943911427197,
1.84340973822864674584708676515, 4.28523033622355814129868253657, 6.42090571719240889596926789405, 7.948190931964489360698003989182, 8.489156025205677787793883593332, 10.34389960238042614711549706495, 12.04541284062115971379734729337, 12.67844185886327658265839141683, 13.82734565746064475153407925710, 14.92539872521718057048154891609
