| L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (0.584 + 0.171i)5-s + (0.415 − 0.909i)6-s + (−0.654 + 0.755i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.398 − 0.460i)10-s + (−4.51 + 2.90i)11-s + (−0.841 + 0.540i)12-s + (−2.05 − 2.37i)13-s + (0.959 − 0.281i)14-s + (−0.0866 + 0.602i)15-s + (−0.654 + 0.755i)16-s + (0.852 − 1.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (0.261 + 0.0767i)5-s + (0.169 − 0.371i)6-s + (−0.247 + 0.285i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.126 − 0.145i)10-s + (−1.36 + 0.874i)11-s + (−0.242 + 0.156i)12-s + (−0.571 − 0.659i)13-s + (0.256 − 0.0752i)14-s + (−0.0223 + 0.155i)15-s + (−0.163 + 0.188i)16-s + (0.206 − 0.452i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.120877 - 0.281730i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.120877 - 0.281730i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (-1.09 + 4.67i)T \) |
| good | 5 | \( 1 + (-0.584 - 0.171i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (4.51 - 2.90i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (2.05 + 2.37i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.852 + 1.86i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (3.09 + 6.78i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.96 - 6.48i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.639 + 4.44i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-7.25 + 2.12i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-10.0 - 2.95i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.483 - 3.36i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 9.08T + 47T^{2} \) |
| 53 | \( 1 + (-2.15 + 2.49i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (0.834 + 0.963i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.625 + 4.35i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (2.95 + 1.89i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (9.80 + 6.30i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.60 + 5.69i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (10.2 + 11.8i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.52 + 0.448i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.01 - 13.9i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.63 - 1.36i)T + (81.6 + 52.4i)T^{2} \) |
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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
−9.653619064228997566782728499397, −9.206664349441707079455112828446, −8.076991868607072646921587846154, −7.45481802571406168693225743802, −6.33340790809260054811952433724, −5.16070349595703704211146445281, −4.43594217188697196751285681310, −2.84076658277097345396553002413, −2.39248535897881748581026972697, −0.16148976357644116665838403095,
1.54240670762341048452279538768, 2.71685157236932728247976672469, 4.08243905258368864121629688018, 5.64367681170650798600114165142, 5.94529896900492486004025126658, 7.17305856335987569369188625638, 7.84892019295629338040406783134, 8.434996495860138399334744019006, 9.537702396927871116009259851375, 10.13876365080912631390686892068
