| L(s) = 1 | + (−0.766 + 0.642i)2-s + (−1.50 − 0.549i)3-s + (0.173 − 0.984i)4-s + (0.142 + 0.810i)5-s + (1.50 − 0.549i)6-s + (−0.0412 + 0.0714i)7-s + (0.500 + 0.866i)8-s + (−0.320 − 0.268i)9-s + (−0.630 − 0.529i)10-s + (0.5 + 0.866i)11-s + (−0.803 + 1.39i)12-s + (0.804 − 0.292i)13-s + (−0.0143 − 0.0812i)14-s + (0.229 − 1.30i)15-s + (−0.939 − 0.342i)16-s + (0.348 − 0.292i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.871 − 0.317i)3-s + (0.0868 − 0.492i)4-s + (0.0639 + 0.362i)5-s + (0.616 − 0.224i)6-s + (−0.0155 + 0.0269i)7-s + (0.176 + 0.306i)8-s + (−0.106 − 0.0895i)9-s + (−0.199 − 0.167i)10-s + (0.150 + 0.261i)11-s + (−0.231 + 0.401i)12-s + (0.223 − 0.0811i)13-s + (−0.00382 − 0.0217i)14-s + (0.0593 − 0.336i)15-s + (−0.234 − 0.0855i)16-s + (0.0844 − 0.0708i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.778610 + 0.0215560i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.778610 + 0.0215560i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-3.98 + 1.77i)T \) |
| good | 3 | \( 1 + (1.50 + 0.549i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.142 - 0.810i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.0412 - 0.0714i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-0.804 + 0.292i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.348 + 0.292i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.705 + 3.99i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.39 - 5.36i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.42 + 2.47i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.05T + 37T^{2} \) |
| 41 | \( 1 + (-8.71 - 3.17i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.99 + 11.2i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (3.22 + 2.70i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.904 - 5.12i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (0.388 - 0.326i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.937 + 5.31i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.16 + 2.65i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.263 - 1.49i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.91 + 0.695i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (6.89 + 2.51i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.501 - 0.868i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.42 - 2.70i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (9.51 - 7.98i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.07970401411631376898578603482, −10.42998351203288267630004771300, −9.350497550043483878777501404204, −8.472214830593684020052842105064, −7.26565270278754099881128813075, −6.58602115387862151601420014829, −5.75044239720067786007037316405, −4.68424188406288628929301904456, −2.86463233106157370817513169778, −0.913880616328701945290512423474,
1.06622882815569855546036256469, 2.92568768601320555350216407532, 4.35626734553235517971830162438, 5.42118911885773552351604364031, 6.37532448421817108405892614300, 7.67862200488785556097012421228, 8.575600337225111438239065314565, 9.586652755183157914529967667289, 10.31759665329636534049249025650, 11.31066616983408048087425898535
