| L(s) = 1 | + (−1.24 + 0.672i)2-s + (−0.949 + 1.33i)3-s + (1.09 − 1.67i)4-s + (0.651 + 3.38i)5-s + (0.283 − 2.29i)6-s + (0.689 − 2.55i)7-s + (−0.234 + 2.81i)8-s + (0.105 + 0.305i)9-s + (−3.08 − 3.76i)10-s + (−3.43 − 4.36i)11-s + (1.19 + 3.04i)12-s + (−2.16 − 3.37i)13-s + (0.861 + 3.64i)14-s + (−5.12 − 2.34i)15-s + (−1.60 − 3.66i)16-s + (−3.31 + 3.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.879 + 0.475i)2-s + (−0.547 + 0.769i)3-s + (0.547 − 0.837i)4-s + (0.291 + 1.51i)5-s + (0.115 − 0.937i)6-s + (0.260 − 0.965i)7-s + (−0.0829 + 0.996i)8-s + (0.0352 + 0.101i)9-s + (−0.976 − 1.19i)10-s + (−1.03 − 1.31i)11-s + (0.344 + 0.879i)12-s + (−0.601 − 0.935i)13-s + (0.230 + 0.973i)14-s + (−1.32 − 0.604i)15-s + (−0.401 − 0.915i)16-s + (−0.804 + 0.843i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0166548 - 0.0203983i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0166548 - 0.0203983i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.24 - 0.672i)T \) |
| 7 | \( 1 + (-0.689 + 2.55i)T \) |
| 23 | \( 1 + (-4.22 + 2.27i)T \) |
| good | 3 | \( 1 + (0.949 - 1.33i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-0.651 - 3.38i)T + (-4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (3.43 + 4.36i)T + (-2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (2.16 + 3.37i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (3.31 - 3.47i)T + (-0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (4.31 - 4.11i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-0.241 + 0.0710i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (5.77 - 0.551i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (2.45 + 7.08i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (3.65 - 3.17i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.538 - 0.246i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-0.0686 + 0.118i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.258 + 5.43i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (4.06 + 2.09i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (0.975 - 0.694i)T + (19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-2.24 - 5.62i)T + (-48.4 + 46.2i)T^{2} \) |
| 71 | \( 1 + (-6.44 - 0.926i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-15.4 + 3.74i)T + (64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (3.45 - 0.164i)T + (78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (2.70 - 3.11i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.0334 + 0.350i)T + (-87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (-5.48 + 4.75i)T + (13.8 - 96.0i)T^{2} \) |
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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
−10.80497105551259724723232774425, −10.54953821288179966299445848003, −9.847419322995286278720477798523, −8.406534211807947605358248764665, −7.71587535680909845891422954538, −6.79733750986855696308041010017, −5.93401944383598505276606936859, −5.10317746026454209040755831963, −3.59439062016494648563866521629, −2.28328756429120445076862357328,
0.01897450956120633094909397780, 1.64175778393070313535659009919, 2.38469833733744385475839299988, 4.57357976259373489368079859111, 5.22357958839267039332951000326, 6.65843752707436004250099478345, 7.31686733528537523949068649422, 8.414087358353594728544668951458, 9.258008583817325549967198506904, 9.518763529544886953259683713741
