L(s) = 1 | + (0.0745 − 0.278i)2-s − 1.06i·3-s + (1.66 + 0.958i)4-s + (0.133 + 0.499i)5-s + (−0.296 − 0.0794i)6-s + (−2.03 − 1.69i)7-s + (0.798 − 0.798i)8-s + 1.86·9-s + 0.148·10-s + (−2.70 + 2.70i)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 + (1.75 + 3.03i)16-s + (−2.26 + 3.91i)17-s + ⋯ |
L(s) = 1 | + (0.0527 − 0.196i)2-s − 0.615i·3-s + (0.830 + 0.479i)4-s + (0.0598 + 0.223i)5-s + (−0.121 − 0.0324i)6-s + (−0.768 − 0.639i)7-s + (0.282 − 0.282i)8-s + 0.621·9-s + 0.0471·10-s + (−0.816 + 0.816i)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 + (0.438 + 0.759i)16-s + (−0.548 + 0.949i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08015 - 0.263141i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08015 - 0.263141i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.03 + 1.69i)T \) |
| 13 | \( 1 + (1.12 + 3.42i)T \) |
good | 2 | \( 1 + (-0.0745 + 0.278i)T + (-1.73 - i)T^{2} \) |
| 3 | \( 1 + 1.06iT - 3T^{2} \) |
| 5 | \( 1 + (-0.133 - 0.499i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (2.70 - 2.70i)T - 11iT^{2} \) |
| 17 | \( 1 + (2.26 - 3.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.17 - 2.17i)T - 19iT^{2} \) |
| 23 | \( 1 + (7.51 - 4.33i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.26 + 2.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.00 - 0.270i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-0.176 - 0.0474i)T + (32.0 + 18.5i)T^{2} \) |
| 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 | \( 1 + (-9.03 + 2.42i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.512 - 0.887i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.503 + 0.134i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + 8.53iT - 61T^{2} \) |
| 67 | \( 1 + (5.00 + 5.00i)T + 67iT^{2} \) |
| 71 | \( 1 + (-1.96 + 7.33i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (3.20 - 11.9i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.77 + 6.54i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.42 - 6.42i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.13 - 11.7i)T + (-77.0 - 44.5i)T^{2} \) |
| 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
−13.67370943263515583481935137177, −12.67619194873878244171622029935, −12.33422184452527717097269280725, −10.56091864684814621427890864945, −10.13462965374452986127460556160, −7.976082111773505152934390615791, −7.23288914303769237847360731004, −6.18800939351104234654502559579, −3.93656970568412780443056793880, −2.23935924684302784037292572617,
2.61878977944804227111654767607, 4.68030504941030942339436599317, 6.03194862804776598648465470939, 7.12259284421450564485882611007, 8.824891294462928812632971523576, 9.894647800691845098374649387160, 10.83220468441591278352386664697, 11.95984963226928509432166040428, 13.14291061357309641002611177083, 14.37000444080213740851548259600