L(s) = 1 | + (0.707 + 0.707i)2-s + (1.63 − 0.566i)3-s + 1.00i·4-s + (0.469 − 1.75i)5-s + (1.55 + 0.757i)6-s + (0.107 − 2.64i)7-s + (−0.707 + 0.707i)8-s + (2.35 − 1.85i)9-s + (1.56 − 0.906i)10-s + (0.871 + 0.233i)11-s + (0.566 + 1.63i)12-s + (−0.922 − 3.48i)13-s + (1.94 − 1.79i)14-s + (−0.223 − 3.13i)15-s − 1.00·16-s − 6.58·17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.945 − 0.326i)3-s + 0.500i·4-s + (0.209 − 0.782i)5-s + (0.635 + 0.309i)6-s + (0.0407 − 0.999i)7-s + (−0.250 + 0.250i)8-s + (0.786 − 0.617i)9-s + (0.496 − 0.286i)10-s + (0.262 + 0.0704i)11-s + (0.163 + 0.472i)12-s + (−0.255 − 0.966i)13-s + (0.519 − 0.479i)14-s + (−0.0576 − 0.808i)15-s − 0.250·16-s − 1.59·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.54646 - 0.432470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.54646 - 0.432470i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-1.63 + 0.566i)T \) |
| 7 | \( 1 + (-0.107 + 2.64i)T \) |
| 13 | \( 1 + (0.922 + 3.48i)T \) |
good | 5 | \( 1 + (-0.469 + 1.75i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.871 - 0.233i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + 6.58T + 17T^{2} \) |
| 19 | \( 1 + (-1.69 - 6.33i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + (-5.47 - 3.16i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.22 - 4.57i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-7.02 - 7.02i)T + 37iT^{2} \) |
| 41 | \( 1 + (12.1 - 3.25i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.99 + 2.30i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.58 - 0.961i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.00 - 2.30i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.79 - 3.79i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.159 - 0.275i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.50 - 0.403i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.215 + 0.803i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-9.62 + 2.57i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.91 + 6.78i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.59 + 7.59i)T + 83iT^{2} \) |
| 89 | \( 1 + (9.27 - 9.27i)T - 89iT^{2} \) |
| 97 | \( 1 + (-8.51 - 2.28i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.57756432235776694877140725095, −9.763956222654202486301915755068, −8.664389373894529410996624645061, −8.116186823496875309565770233402, −7.16956911117575903796782169233, −6.34967856092832493054140967744, −4.95731723291269193150448385423, −4.11856259222445098336586656684, −3.01227445819211983709119311049, −1.35611098670661501993863751837,
2.26329577594396285925613909501, 2.64609747120776404021464052375, 4.06621708045574388819583280993, 4.91587666529700592016789857850, 6.35403089720459495651794316487, 7.05934106394350826562342806300, 8.514982871700154847009227416833, 9.193824957461933297105064410873, 9.869545402283377455405460603789, 11.00879188362556907161771190678