| L(s) = 1 | + (0.470 + 1.33i)2-s + (−1.72 + 0.0988i)3-s + (−1.55 + 1.25i)4-s + (−0.580 + 0.691i)5-s + (−0.944 − 2.25i)6-s + (−1.54 + 4.24i)7-s + (−2.40 − 1.48i)8-s + (2.98 − 0.341i)9-s + (−1.19 − 0.448i)10-s + (3.82 − 3.20i)11-s + (2.57 − 2.32i)12-s + (−0.531 + 3.01i)13-s + (−6.38 − 0.0650i)14-s + (0.935 − 1.25i)15-s + (0.854 − 3.90i)16-s + (1.45 + 0.841i)17-s + ⋯ |
| L(s) = 1 | + (0.332 + 0.943i)2-s + (−0.998 + 0.0570i)3-s + (−0.778 + 0.627i)4-s + (−0.259 + 0.309i)5-s + (−0.385 − 0.922i)6-s + (−0.583 + 1.60i)7-s + (−0.850 − 0.526i)8-s + (0.993 − 0.113i)9-s + (−0.377 − 0.141i)10-s + (1.15 − 0.967i)11-s + (0.741 − 0.670i)12-s + (−0.147 + 0.836i)13-s + (−1.70 − 0.0173i)14-s + (0.241 − 0.323i)15-s + (0.213 − 0.976i)16-s + (0.353 + 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.245741 + 0.713260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.245741 + 0.713260i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.470 - 1.33i)T \) |
| 3 | \( 1 + (1.72 - 0.0988i)T \) |
| good | 5 | \( 1 + (0.580 - 0.691i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.54 - 4.24i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-3.82 + 3.20i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.531 - 3.01i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.45 - 0.841i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.58 + 1.49i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.595 + 0.216i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (3.15 - 0.556i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.74 - 4.79i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.12 + 5.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.604 + 0.106i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (2.73 + 3.26i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.594 - 0.216i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 9.93iT - 53T^{2} \) |
| 59 | \( 1 + (-5.76 - 4.83i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.30 + 0.838i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-6.52 - 1.15i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.592 - 1.02i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.03 - 3.51i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.274 - 0.0484i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.794 + 4.50i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-11.5 + 6.69i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.79 - 4.02i)T + (16.8 - 95.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.34414589172469093453011365382, −13.01161816053426170716107499550, −12.01305217081586241979468832571, −11.43627903589622382976053956580, −9.533880850908765175188056772794, −8.762292405914824296108941623666, −7.02998795456120088715080300704, −6.16430201991341459445022246566, −5.27742698717771073839387358122, −3.58104319371805037533816351560,
0.966111655821775479981238858745, 3.80195799245331752630549602571, 4.74012130226940257346005783323, 6.32386568597768957470100055511, 7.59036733257041617908964163415, 9.685372849983069230950473864818, 10.17957984208578901320871708993, 11.31741207465285009591578868788, 12.23549299378266338328220988756, 12.99307038875407690378918223136
