| L(s) = 1 | + (−2.20 − 1.54i)2-s + (−1.18 − 1.26i)3-s + (1.78 + 4.91i)4-s + (0.304 + 3.47i)5-s + (0.654 + 4.60i)6-s + (−0.273 − 0.229i)7-s + (2.24 − 8.37i)8-s + (−0.200 + 2.99i)9-s + (4.69 − 8.12i)10-s + (0.341 + 0.591i)11-s + (4.09 − 8.06i)12-s + (−0.409 + 0.879i)13-s + (0.248 + 0.926i)14-s + (4.03 − 4.49i)15-s + (−9.84 + 8.26i)16-s + (1.96 + 4.20i)17-s + ⋯ |
| L(s) = 1 | + (−1.55 − 1.09i)2-s + (−0.683 − 0.730i)3-s + (0.893 + 2.45i)4-s + (0.136 + 1.55i)5-s + (0.267 + 1.88i)6-s + (−0.103 − 0.0867i)7-s + (0.793 − 2.96i)8-s + (−0.0669 + 0.997i)9-s + (1.48 − 2.56i)10-s + (0.102 + 0.178i)11-s + (1.18 − 2.32i)12-s + (−0.113 + 0.243i)13-s + (0.0663 + 0.247i)14-s + (1.04 − 1.16i)15-s + (−2.46 + 2.06i)16-s + (0.475 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.371593 + 0.0354742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.371593 + 0.0354742i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.18 + 1.26i)T \) |
| 37 | \( 1 + (2.11 - 5.70i)T \) |
| good | 2 | \( 1 + (2.20 + 1.54i)T + (0.684 + 1.87i)T^{2} \) |
| 5 | \( 1 + (-0.304 - 3.47i)T + (-4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (0.273 + 0.229i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.341 - 0.591i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.409 - 0.879i)T + (-8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-1.96 - 4.20i)T + (-10.9 + 13.0i)T^{2} \) |
| 19 | \( 1 + (-2.50 - 3.58i)T + (-6.49 + 17.8i)T^{2} \) |
| 23 | \( 1 + (0.336 - 0.0902i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.92 - 0.782i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (2.48 + 2.48i)T + 31iT^{2} \) |
| 41 | \( 1 + (1.60 - 0.584i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (4.75 - 4.75i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.43 - 1.40i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.77 + 5.69i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-3.27 - 0.286i)T + (58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 4.83i)T + (39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-5.06 + 6.03i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (14.8 + 2.62i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + 3.02iT - 73T^{2} \) |
| 79 | \( 1 + (-10.3 + 0.904i)T + (77.7 - 13.7i)T^{2} \) |
| 83 | \( 1 + (-3.62 + 9.94i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.625 + 7.14i)T + (-87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (3.14 + 11.7i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−13.23661213429298800870634371293, −12.08407744845823063547860935178, −11.40727089335087742302236591164, −10.45361656605010351008703567076, −9.939711986278357303025296144851, −8.211314461875964037745432178525, −7.29508810024306074300292681903, −6.37328191183750658124377782738, −3.32167120294357878219022507019, −1.85310621823144622119938120146,
0.77263293389670675705700554062, 4.96431822242160302317471371355, 5.66448929339819246387922271133, 7.09302709815753941948460767691, 8.495113884806319629475841797367, 9.253946429661419458368991099615, 9.913919436955682363778375303452, 11.16385060753228593342291232775, 12.29597283438163095834648238622, 13.96515453721098601570875728750
