| L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.415 + 0.909i)6-s + (0.988 + 1.14i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)10-s + (−3.77 − 2.42i)11-s + (−0.841 − 0.540i)12-s + (1.59 − 1.83i)13-s + (−1.44 − 0.425i)14-s + (0.142 + 0.989i)15-s + (−0.654 − 0.755i)16-s + (−0.823 − 1.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (−0.429 + 0.125i)5-s + (0.169 + 0.371i)6-s + (0.373 + 0.431i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.207 − 0.238i)10-s + (−1.13 − 0.732i)11-s + (−0.242 − 0.156i)12-s + (0.441 − 0.509i)13-s + (−0.386 − 0.113i)14-s + (0.0367 + 0.255i)15-s + (−0.163 − 0.188i)16-s + (−0.199 − 0.437i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.333959 - 0.519229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.333959 - 0.519229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (0.877 + 4.71i)T \) |
| good | 7 | \( 1 + (-0.988 - 1.14i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (3.77 + 2.42i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.59 + 1.83i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.823 + 1.80i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.110 - 0.242i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.31 + 2.87i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.838 - 5.82i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (11.2 + 3.29i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-7.39 + 2.17i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.74 + 12.1i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 9.15T + 47T^{2} \) |
| 53 | \( 1 + (8.83 + 10.1i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (9.60 - 11.0i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.873 - 6.07i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-0.869 + 0.558i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.47 + 3.52i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.679 - 1.48i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-5.79 + 6.68i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-7.15 - 2.10i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.948 + 6.59i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-11.4 + 3.36i)T + (81.6 - 52.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.37849763108855992851871174529, −8.950813662878970116021006124606, −8.394575170666932936613936123223, −7.72966784591653418641019693238, −6.83492544662907026066434604056, −5.81375742214845108184896116189, −5.00516146629543976489848551449, −3.31635674762674377775740560084, −2.14159972629536872508305380155, −0.38195091221793221313559138846,
1.68977208842123459148366859199, 3.12248274505666356982825127755, 4.21353615685516952012761520478, 5.04100352171374232395045970665, 6.43632638381094528285277621387, 7.71606363550951961099061338039, 8.025794635648776930556990852872, 9.236620538451265258068231599729, 9.814675070399519473760282934004, 10.91768628097270546232899281282
