| L(s) = 1 | + (−1.28 − 0.584i)2-s + (−1.68 − 0.388i)3-s + (1.31 + 1.50i)4-s + (−0.837 + 0.382i)5-s + (1.94 + 1.48i)6-s + (3.05 + 3.20i)7-s + (−0.817 − 2.70i)8-s + (2.69 + 1.31i)9-s + (1.30 − 0.00322i)10-s + (2.13 + 0.203i)11-s + (−1.63 − 3.05i)12-s + (−2.08 − 0.401i)13-s + (−2.06 − 5.91i)14-s + (1.56 − 0.320i)15-s + (−0.530 + 3.96i)16-s + (−1.63 + 3.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.910 − 0.413i)2-s + (−0.974 − 0.224i)3-s + (0.658 + 0.752i)4-s + (−0.374 + 0.171i)5-s + (0.794 + 0.606i)6-s + (1.15 + 1.21i)7-s + (−0.288 − 0.957i)8-s + (0.899 + 0.436i)9-s + (0.411 − 0.00101i)10-s + (0.643 + 0.0614i)11-s + (−0.473 − 0.880i)12-s + (−0.577 − 0.111i)13-s + (−0.551 − 1.58i)14-s + (0.403 − 0.0827i)15-s + (−0.132 + 0.991i)16-s + (−0.396 + 0.768i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.497906 + 0.382087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.497906 + 0.382087i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 + 0.584i)T \) |
| 3 | \( 1 + (1.68 + 0.388i)T \) |
| 67 | \( 1 + (6.69 + 4.70i)T \) |
| good | 5 | \( 1 + (0.837 - 0.382i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-3.05 - 3.20i)T + (-0.333 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-2.13 - 0.203i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (2.08 + 0.401i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (1.63 - 3.16i)T + (-9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (-2.30 + 2.42i)T + (-0.904 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-2.93 + 1.17i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (8.62 + 4.97i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.08 - 5.64i)T + (-28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (0.0290 + 0.0503i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.16 + 0.150i)T + (40.8 - 3.89i)T^{2} \) |
| 43 | \( 1 + (-2.25 - 3.51i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (0.135 + 0.106i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (5.17 - 8.05i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-8.26 - 9.54i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-11.0 + 1.05i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (-1.27 + 0.656i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (0.405 - 0.0387i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (2.72 - 0.942i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (5.41 - 7.60i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (5.56 - 0.800i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (2.40 + 4.16i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.66770174160816269687040013909, −9.544469591521827318462095653388, −8.840882457272638544467381082775, −7.85148176825291355042440999801, −7.21681285937514466685108937183, −6.15083988038845659635755813117, −5.20119041670549828874764736785, −4.05895901139607207800047707849, −2.43607524628880958718609541776, −1.38313509891542970207940140775,
0.52685021365299458553976106443, 1.69423438121772730336614173260, 3.93227835752979757124752954595, 4.85464208693455858207315069079, 5.68572001753739504237952344802, 6.99561515181082082897716964287, 7.32905119910139941602983470889, 8.245542412560506250543358424938, 9.427443336595309375675986966666, 10.00677198262974451879917580415
