L(s) = 1 | + (0.203 + 0.203i)2-s + (0.923 − 0.532i)3-s − 1.91i·4-s + (0.499 + 0.133i)5-s + (0.296 + 0.0794i)6-s + (−2.60 + 0.449i)7-s + (0.798 − 0.798i)8-s + (−0.931 + 1.61i)9-s + (0.0744 + 0.129i)10-s + (3.69 + 0.990i)11-s + (−1.02 − 1.76i)12-s + (1.12 + 3.42i)13-s + (−0.622 − 0.439i)14-s + (0.532 − 0.142i)15-s − 3.50·16-s − 4.52·17-s + ⋯ |
L(s) = 1 | + (0.144 + 0.144i)2-s + (0.532 − 0.307i)3-s − 0.958i·4-s + (0.223 + 0.0598i)5-s + (0.121 + 0.0324i)6-s + (−0.985 + 0.169i)7-s + (0.282 − 0.282i)8-s + (−0.310 + 0.538i)9-s + (0.0235 + 0.0407i)10-s + (1.11 + 0.298i)11-s + (−0.294 − 0.510i)12-s + (0.312 + 0.950i)13-s + (−0.166 − 0.117i)14-s + (0.137 − 0.0368i)15-s − 0.877·16-s − 1.09·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14156 - 0.235387i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14156 - 0.235387i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.60 - 0.449i)T \) |
| 13 | \( 1 + (-1.12 - 3.42i)T \) |
good | 2 | \( 1 + (-0.203 - 0.203i)T + 2iT^{2} \) |
| 3 | \( 1 + (-0.923 + 0.532i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.499 - 0.133i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-3.69 - 0.990i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + (-0.797 - 2.97i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + 8.67iT - 23T^{2} \) |
| 29 | \( 1 + (-1.26 + 2.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.270 - 1.00i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (0.129 - 0.129i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.79 - 6.68i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-4.59 + 2.65i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.42 + 9.03i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.512 + 0.887i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.368 - 0.368i)T + 59iT^{2} \) |
| 61 | \( 1 + (7.39 + 4.26i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.83 - 6.83i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.96 + 7.33i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (11.9 - 3.20i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.77 - 6.54i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.42 + 6.42i)T - 83iT^{2} \) |
| 89 | \( 1 + (-8.56 - 8.56i)T + 89iT^{2} \) |
| 97 | \( 1 + (13.1 + 3.53i)T + (84.0 + 48.5i)T^{2} \) |
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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
−14.04261827133941753482021707706, −13.33000262686259053994620103681, −11.97532597705304311654030299442, −10.65759176641612815772786067387, −9.548748373263323028483321344215, −8.678503104721378123214211404581, −6.82710473650440435389575365207, −6.11317851296678870866973094152, −4.30711412896718686583683432710, −2.15767277195747218401145427645,
3.05660061842406579729998440505, 3.95021359306661374594861742803, 6.09056727341958373996318646608, 7.41871832234766103840719664920, 8.899298407630737525908274581386, 9.431590495115686760887900322132, 11.09460282990907380999085578177, 12.13216827856725074788691976320, 13.25078610583381281405747036353, 13.85222820791959294706771086819