L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.40 − 1.01i)3-s + 1.00i·4-s + (2.22 + 0.198i)5-s + (−1.71 − 0.275i)6-s + (0.787 − 0.787i)7-s + (0.707 − 0.707i)8-s + (0.940 − 2.84i)9-s + (−1.43 − 1.71i)10-s + 0.307i·11-s + (1.01 + 1.40i)12-s + (−4.92 − 4.92i)13-s − 1.11·14-s + (3.32 − 1.98i)15-s − 1.00·16-s + (0.790 + 0.790i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.810 − 0.585i)3-s + 0.500i·4-s + (0.996 + 0.0889i)5-s + (−0.698 − 0.112i)6-s + (0.297 − 0.297i)7-s + (0.250 − 0.250i)8-s + (0.313 − 0.949i)9-s + (−0.453 − 0.542i)10-s + 0.0926i·11-s + (0.292 + 0.405i)12-s + (−1.36 − 1.36i)13-s − 0.297·14-s + (0.859 − 0.511i)15-s − 0.250·16-s + (0.191 + 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0167 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0167 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25367 - 1.27482i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25367 - 1.27482i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.40 + 1.01i)T \) |
| 5 | \( 1 + (-2.22 - 0.198i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-0.787 + 0.787i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.307iT - 11T^{2} \) |
| 13 | \( 1 + (4.92 + 4.92i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.790 - 0.790i)T + 17iT^{2} \) |
| 19 | \( 1 - 0.691iT - 19T^{2} \) |
| 29 | \( 1 - 6.67T + 29T^{2} \) |
| 31 | \( 1 - 4.52T + 31T^{2} \) |
| 37 | \( 1 + (-1.77 + 1.77i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.596iT - 41T^{2} \) |
| 43 | \( 1 + (6.70 + 6.70i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.88 - 6.88i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.94 - 5.94i)T - 53iT^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 + 9.95T + 61T^{2} \) |
| 67 | \( 1 + (-2.19 + 2.19i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.55iT - 71T^{2} \) |
| 73 | \( 1 + (4.90 + 4.90i)T + 73iT^{2} \) |
| 79 | \( 1 - 15.7iT - 79T^{2} \) |
| 83 | \( 1 + (-6.65 + 6.65i)T - 83iT^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + (12.8 - 12.8i)T - 97iT^{2} \) |
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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
−10.05103045823422688538742128100, −9.571174788524427594915993229374, −8.510652414530118231625081522413, −7.79109839187602436648072738018, −7.01309363983914524397216879634, −5.88323843920781771312092574248, −4.62133566943057512681533894126, −3.08490975733308009575788374630, −2.37389083488768756759210251658, −1.09703947836175838073697618833,
1.82826128190255722060684597169, 2.75534433780118905964781399794, 4.53643115647868616239392165282, 5.11726803934707920524541909789, 6.40016653377770503006415888325, 7.24518597677552007303228146606, 8.355853231174019730782323316503, 8.956308634982898142915094208443, 9.867521196752695269950876945912, 10.05877681176817459998767879574