L(s) = 1 | + (−0.499 − 1.32i)2-s + (3.11 + 0.833i)3-s + (−1.50 + 1.32i)4-s + (−1.44 − 1.70i)5-s + (−0.450 − 4.53i)6-s + (2.23 + 1.42i)7-s + (2.49 + 1.32i)8-s + (6.39 + 3.68i)9-s + (−1.53 + 2.76i)10-s + (−2.06 − 3.57i)11-s + (−5.77 + 2.85i)12-s + (1.73 + 1.73i)13-s + (0.770 − 3.66i)14-s + (−3.07 − 6.51i)15-s + (0.510 − 3.96i)16-s + (0.356 − 1.32i)17-s + ⋯ |
L(s) = 1 | + (−0.352 − 0.935i)2-s + (1.79 + 0.481i)3-s + (−0.750 + 0.660i)4-s + (−0.646 − 0.763i)5-s + (−0.183 − 1.85i)6-s + (0.842 + 0.538i)7-s + (0.882 + 0.469i)8-s + (2.13 + 1.22i)9-s + (−0.485 + 0.874i)10-s + (−0.622 − 1.07i)11-s + (−1.66 + 0.825i)12-s + (0.482 + 0.482i)13-s + (0.205 − 0.978i)14-s + (−0.793 − 1.68i)15-s + (0.127 − 0.991i)16-s + (0.0863 − 0.322i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57415 - 0.651037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57415 - 0.651037i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.499 + 1.32i)T \) |
| 5 | \( 1 + (1.44 + 1.70i)T \) |
| 7 | \( 1 + (-2.23 - 1.42i)T \) |
good | 3 | \( 1 + (-3.11 - 0.833i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (2.06 + 3.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.356 + 1.32i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.51 + 0.872i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.71 - 0.994i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 0.469T + 29T^{2} \) |
| 31 | \( 1 + (-0.329 + 0.190i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.20 + 4.48i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 + (5.07 - 5.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.284 + 1.06i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.00467 - 0.0174i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (7.49 - 4.32i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.35 + 3.66i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.65 + 9.92i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 + (-7.60 - 2.03i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (3.90 - 6.76i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.47 + 2.47i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.00 - 3.46i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.49 + 8.49i)T + 97iT^{2} \) |
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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
−11.63375537741730526081561669234, −10.76501196922541777871529328965, −9.593940253410357050373186058480, −8.741345885464355548733331301060, −8.347748125102652552085795681228, −7.68523916753716942502111262883, −5.03859547486689022523326056207, −4.05716365275011967470246613547, −3.07860686306565646249935472454, −1.77622048065076168495719598698,
1.87305660579739929382478615819, 3.55235172372003887197862894935, 4.60785474977520283305934962054, 6.59937861216653640700162313693, 7.47040811510517322722694052347, 7.997848127831512664612933485485, 8.591720982131685291529017773538, 9.970530924223154098106013754111, 10.53547306413900109312168643507, 12.25281787066508898316583969460