| L(s) = 1 | + (0.473 + 1.33i)2-s + (1.50 + 1.12i)3-s + (−1.55 + 1.26i)4-s + (−1.88 + 1.19i)5-s + (−0.785 + 2.53i)6-s + (−1.48 + 0.105i)7-s + (−2.41 − 1.46i)8-s + (0.144 + 0.493i)9-s + (−2.48 − 1.95i)10-s + (−1.25 − 1.95i)11-s + (−3.74 + 0.153i)12-s + (−0.414 + 5.79i)13-s + (−0.843 − 1.92i)14-s + (−4.17 − 0.329i)15-s + (0.809 − 3.91i)16-s + (0.0539 + 0.144i)17-s + ⋯ |
| L(s) = 1 | + (0.335 + 0.942i)2-s + (0.866 + 0.648i)3-s + (−0.775 + 0.631i)4-s + (−0.845 + 0.534i)5-s + (−0.320 + 1.03i)6-s + (−0.560 + 0.0400i)7-s + (−0.854 − 0.518i)8-s + (0.0482 + 0.164i)9-s + (−0.786 − 0.617i)10-s + (−0.379 − 0.590i)11-s + (−1.08 + 0.0442i)12-s + (−0.114 + 1.60i)13-s + (−0.225 − 0.514i)14-s + (−1.07 − 0.0851i)15-s + (0.202 − 0.979i)16-s + (0.0130 + 0.0350i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.117134 - 1.30922i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.117134 - 1.30922i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.473 - 1.33i)T \) |
| 5 | \( 1 + (1.88 - 1.19i)T \) |
| 23 | \( 1 + (4.75 + 0.609i)T \) |
| good | 3 | \( 1 + (-1.50 - 1.12i)T + (0.845 + 2.87i)T^{2} \) |
| 7 | \( 1 + (1.48 - 0.105i)T + (6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (1.25 + 1.95i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.414 - 5.79i)T + (-12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-0.0539 - 0.144i)T + (-12.8 + 11.1i)T^{2} \) |
| 19 | \( 1 + (-3.30 - 7.22i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-9.44 - 4.31i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.25 - 0.179i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (4.79 - 8.79i)T + (-20.0 - 31.1i)T^{2} \) |
| 41 | \( 1 + (-7.19 - 2.11i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (4.62 + 3.46i)T + (12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-1.34 - 1.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.99 + 0.500i)T + (52.4 - 7.54i)T^{2} \) |
| 59 | \( 1 + (-1.35 - 1.56i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.158 - 1.10i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (2.96 + 13.6i)T + (-60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-5.23 + 8.13i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (4.32 - 11.6i)T + (-55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (4.74 + 5.47i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (4.65 - 8.52i)T + (-44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-8.92 + 1.28i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (3.14 - 1.71i)T + (52.4 - 81.6i)T^{2} \) |
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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
−11.86125388077623051566886513267, −10.30777641637280666067544614743, −9.583517943604810204146986550814, −8.547964569595878693394184627493, −8.053155414319393670788815145009, −6.87971945469626723561369756189, −6.13122669340490892251644234356, −4.59374775439360469867867553874, −3.72393001765086767583273284621, −3.02734722403969649939963266004,
0.66596323504579774662475401146, 2.48347644930809299366640459074, 3.24628358987982653016175668237, 4.53814849951422396792736057460, 5.53102509021247690697380102397, 7.15925410057602656418541895486, 8.024740312664992685122498773085, 8.744089526983928516573756667308, 9.760404451576084206124710032979, 10.60655458868447762828319879049
