L(s) = 1 | + (0.794 + 0.458i)2-s + (−3.57 − 6.19i)4-s + 15.4i·5-s + (17.8 − 10.2i)7-s − 13.9i·8-s + (−7.09 + 12.2i)10-s + (57.0 + 32.9i)11-s + (19.2 + 42.7i)13-s + 18.8·14-s + (−22.2 + 38.5i)16-s + (−22.1 − 38.3i)17-s + (127. − 73.5i)19-s + (95.9 − 55.3i)20-s + (30.2 + 52.3i)22-s + (−26.5 + 46.0i)23-s + ⋯ |
L(s) = 1 | + (0.280 + 0.162i)2-s + (−0.447 − 0.774i)4-s + 1.38i·5-s + (0.962 − 0.555i)7-s − 0.614i·8-s + (−0.224 + 0.388i)10-s + (1.56 + 0.902i)11-s + (0.410 + 0.911i)13-s + 0.360·14-s + (−0.347 + 0.602i)16-s + (−0.315 − 0.547i)17-s + (1.53 − 0.887i)19-s + (1.07 − 0.619i)20-s + (0.292 + 0.506i)22-s + (−0.240 + 0.417i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.89910 + 0.420560i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.89910 + 0.420560i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (-19.2 - 42.7i)T \) |
good | 2 | \( 1 + (-0.794 - 0.458i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 15.4iT - 125T^{2} \) |
| 7 | \( 1 + (-17.8 + 10.2i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-57.0 - 32.9i)T + (665.5 + 1.15e3i)T^{2} \) |
| 17 | \( 1 + (22.1 + 38.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-127. + 73.5i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (26.5 - 46.0i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (19.3 - 33.4i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 88.3iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-68.3 - 39.4i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (307. + 177. i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (203. + 353. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 67.9iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 226.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-123. + 71.0i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (133. + 231. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-356. - 205. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-79.2 + 45.7i)T + (1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + 63.1iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 287.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 373. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (103. + 59.7i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-480. + 277. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.78095059889836858943815941766, −11.77609541734272372095468493940, −11.13385637267798234724210777387, −9.981652784309459153664440721937, −9.083443173977975232363146332417, −7.15360097545657682425968391316, −6.66499234857858236127867815930, −4.97478511427926818972355170378, −3.78767934683981654292035265483, −1.55615348120379106292653232142,
1.26245744090844475105604671636, 3.56618314866364362492282010876, 4.76838868846152978875898021096, 5.84613543060394813465187043530, 8.063685590066038915066592853400, 8.489075147010780920092029148894, 9.466091164858474910083942093318, 11.42921653479934102057874593895, 11.96509489497213399006040753947, 12.90686358104427131210197075173