| L(s) = 1 | + (1.23 − 3.38i)2-s + (3.55 − 0.626i)3-s + (−6.89 − 5.78i)4-s + (3.06 − 2.57i)5-s + (2.25 − 12.8i)6-s + (2.5 − 4.33i)7-s + (−15.6 + 9.01i)8-s + (3.75 − 1.36i)9-s + (−4.93 − 13.5i)10-s + (5 + 8.66i)11-s + (−28.1 − 16.2i)12-s + (3.55 + 0.626i)13-s + (−11.5 − 13.8i)14-s + (9.27 − 11.0i)15-s + (5.03 + 28.5i)16-s + (−14.0 − 5.13i)17-s + ⋯ |
| L(s) = 1 | + (0.616 − 1.69i)2-s + (1.18 − 0.208i)3-s + (−1.72 − 1.44i)4-s + (0.612 − 0.514i)5-s + (0.376 − 2.13i)6-s + (0.357 − 0.618i)7-s + (−1.95 + 1.12i)8-s + (0.417 − 0.152i)9-s + (−0.493 − 1.35i)10-s + (0.454 + 0.787i)11-s + (−2.34 − 1.35i)12-s + (0.273 + 0.0481i)13-s + (−0.827 − 0.986i)14-s + (0.618 − 0.736i)15-s + (0.314 + 1.78i)16-s + (−0.829 − 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.228949 - 3.37265i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228949 - 3.37265i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-1.23 + 3.38i)T + (-3.06 - 2.57i)T^{2} \) |
| 3 | \( 1 + (-3.55 + 0.626i)T + (8.45 - 3.07i)T^{2} \) |
| 5 | \( 1 + (-3.06 + 2.57i)T + (4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-2.5 + 4.33i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-5 - 8.66i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.55 - 0.626i)T + (158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (14.0 + 5.13i)T + (221. + 185. i)T^{2} \) |
| 23 | \( 1 + (-26.8 - 22.4i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (6.16 + 16.9i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-31.2 - 18.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 21.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (35.5 - 6.26i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (15.3 - 12.8i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (9.39 - 3.42i)T + (1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-48.6 + 58.0i)T + (-487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-6.16 + 16.9i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (30.6 + 25.7i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (13.5 + 37.2i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-69.5 - 82.8i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-18.2 - 103. i)T + (-5.00e3 + 1.82e3i)T^{2} \) |
| 79 | \( 1 + (-35.5 + 6.26i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-20 + 34.6i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (-41.9 + 115. i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−10.94675043460327501394893113598, −9.777104958755052216315060448224, −9.318131113031225297487621154174, −8.422451849115402447451127358469, −7.04677355340511889486692567315, −5.30220818811031376378801939122, −4.35590070705764170177002090562, −3.33003870149054284475342798454, −2.15778171514156661703795885353, −1.29632702582374403046409937682,
2.57934587492784271410849558165, 3.72357456935615508654174647022, 4.93037917427175197183228116418, 6.09649484923640994183501266718, 6.70051365305177214787949674307, 7.978927626894909758687126522193, 8.720227186658936141174843739498, 9.096218718032077664582507028434, 10.54475687175013156784728707434, 11.91965550620052087063403794566
