L(s) = 1 | + 0.274i·2-s + (1.67 − 1.67i)3-s + 1.92·4-s + (1.45 + 1.69i)5-s + (0.459 + 0.459i)6-s + 0.386·7-s + 1.07i·8-s − 2.58i·9-s + (−0.466 + 0.399i)10-s + (−3.08 + 3.08i)11-s + (3.21 − 3.21i)12-s + 0.106i·14-s + (5.26 + 0.409i)15-s + 3.55·16-s + (−1.39 + 1.39i)17-s + 0.710·18-s + ⋯ |
L(s) = 1 | + 0.194i·2-s + (0.964 − 0.964i)3-s + 0.962·4-s + (0.650 + 0.759i)5-s + (0.187 + 0.187i)6-s + 0.145·7-s + 0.381i·8-s − 0.861i·9-s + (−0.147 + 0.126i)10-s + (−0.929 + 0.929i)11-s + (0.928 − 0.928i)12-s + 0.0283i·14-s + (1.36 + 0.105i)15-s + 0.888·16-s + (−0.338 + 0.338i)17-s + 0.167·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.88989 - 0.0221645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.88989 - 0.0221645i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.45 - 1.69i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.274iT - 2T^{2} \) |
| 3 | \( 1 + (-1.67 + 1.67i)T - 3iT^{2} \) |
| 7 | \( 1 - 0.386T + 7T^{2} \) |
| 11 | \( 1 + (3.08 - 3.08i)T - 11iT^{2} \) |
| 17 | \( 1 + (1.39 - 1.39i)T - 17iT^{2} \) |
| 19 | \( 1 + (-3.54 + 3.54i)T - 19iT^{2} \) |
| 23 | \( 1 + (0.235 + 0.235i)T + 23iT^{2} \) |
| 29 | \( 1 + 8.16iT - 29T^{2} \) |
| 31 | \( 1 + (2.54 + 2.54i)T + 31iT^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 + (3.29 + 3.29i)T + 41iT^{2} \) |
| 43 | \( 1 + (4.82 + 4.82i)T + 43iT^{2} \) |
| 47 | \( 1 + 9.83T + 47T^{2} \) |
| 53 | \( 1 + (7.17 - 7.17i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.71 + 1.71i)T + 59iT^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 6.37iT - 67T^{2} \) |
| 71 | \( 1 + (3.07 + 3.07i)T + 71iT^{2} \) |
| 73 | \( 1 + 6.08iT - 73T^{2} \) |
| 79 | \( 1 + 3.34iT - 79T^{2} \) |
| 83 | \( 1 - 5.18T + 83T^{2} \) |
| 89 | \( 1 + (3.53 + 3.53i)T + 89iT^{2} \) |
| 97 | \( 1 - 14.7iT - 97T^{2} \) |
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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
−10.15121760370021609588218053593, −9.372440719181474994592560724719, −8.077429191262691468732138574962, −7.60757875466253561027510628860, −6.90538788343341516028088137002, −6.19851799440128426012392063050, −5.04117589408957876972175099942, −3.23446939484160125320823040752, −2.38344948395518168897623577202, −1.82649452496407735372914830778,
1.52816204998040045852550528471, 2.82585707681767409687974846974, 3.46226279255571799327655221384, 4.85921910199126947393599984166, 5.61614223267892496817075050067, 6.73028041454928267556159711232, 8.053752333057241209463219276928, 8.453712234126871896950602116560, 9.577333865967520718191297516225, 9.994063071058615263260426453591