L(s) = 1 | + (0.0473 − 0.176i)2-s + (−0.866 + 0.5i)3-s + (1.70 + 0.983i)4-s + (2.80 − 2.80i)5-s + (0.0473 + 0.176i)6-s + (−0.467 + 2.60i)7-s + (0.513 − 0.513i)8-s + (0.499 − 0.866i)9-s + (−0.362 − 0.627i)10-s + (−2.53 − 0.679i)11-s − 1.96·12-s + (1.37 − 3.33i)13-s + (0.437 + 0.205i)14-s + (−1.02 + 3.82i)15-s + (1.90 + 3.29i)16-s + (−1.43 + 2.48i)17-s + ⋯ |
L(s) = 1 | + (0.0334 − 0.124i)2-s + (−0.499 + 0.288i)3-s + (0.851 + 0.491i)4-s + (1.25 − 1.25i)5-s + (0.0193 + 0.0721i)6-s + (−0.176 + 0.984i)7-s + (0.181 − 0.181i)8-s + (0.166 − 0.288i)9-s + (−0.114 − 0.198i)10-s + (−0.764 − 0.204i)11-s − 0.567·12-s + (0.380 − 0.924i)13-s + (0.117 + 0.0550i)14-s + (−0.264 + 0.987i)15-s + (0.475 + 0.822i)16-s + (−0.347 + 0.601i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55037 - 0.00419139i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55037 - 0.00419139i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.467 - 2.60i)T \) |
| 13 | \( 1 + (-1.37 + 3.33i)T \) |
good | 2 | \( 1 + (-0.0473 + 0.176i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (-2.80 + 2.80i)T - 5iT^{2} \) |
| 11 | \( 1 + (2.53 + 0.679i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (1.43 - 2.48i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.759 - 2.83i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-7.27 + 4.19i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.66 - 2.87i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.75 - 6.75i)T - 31iT^{2} \) |
| 37 | \( 1 + (6.77 + 1.81i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.79 + 0.747i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (2.43 + 1.40i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.85 + 4.85i)T + 47iT^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 + (0.00666 - 0.00178i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-5.65 - 3.26i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.10 + 7.84i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (14.5 - 3.91i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.321 - 0.321i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.280T + 79T^{2} \) |
| 83 | \( 1 + (2.42 - 2.42i)T - 83iT^{2} \) |
| 89 | \( 1 + (-0.0536 + 0.200i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.197 - 0.736i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−12.15547658261281310256609257668, −10.82372021295851859767672554953, −10.22675505367413257982276501764, −8.914197118755909177339282513536, −8.360846683320718013382705570916, −6.69125989108883082314353096419, −5.65266425166904027927764755142, −5.11224585800308918437566961066, −3.13975699990082171899387736367, −1.71420286944366368372770193976,
1.74336099782161340308973105049, 2.98802421535780030892117527889, 5.06454220199130085377892318517, 6.13740770942159434803705945743, 6.95038621502901646081820443077, 7.33466039191884260834465430848, 9.473332416415418969966733915605, 10.18250398761848916507529556151, 11.09641184931661417010751978581, 11.34213334141124735281674727028