L(s) = 1 | + (2.16 + 2.16i)3-s + (−1.91 + 1.14i)5-s + (−2.61 − 0.409i)7-s + 6.36i·9-s + 0.796·11-s + (3.25 + 3.25i)13-s + (−6.63 − 1.66i)15-s + (2.52 − 2.52i)17-s + 2.29·19-s + (−4.76 − 6.54i)21-s + (−2.08 + 2.08i)23-s + (2.35 − 4.40i)25-s + (−7.27 + 7.27i)27-s − 10.0i·29-s − 3.63i·31-s + ⋯ |
L(s) = 1 | + (1.24 + 1.24i)3-s + (−0.857 + 0.513i)5-s + (−0.987 − 0.154i)7-s + 2.12i·9-s + 0.240·11-s + (0.901 + 0.901i)13-s + (−1.71 − 0.429i)15-s + (0.611 − 0.611i)17-s + 0.527·19-s + (−1.04 − 1.42i)21-s + (−0.434 + 0.434i)23-s + (0.471 − 0.881i)25-s + (−1.39 + 1.39i)27-s − 1.87i·29-s − 0.652i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.996431 + 1.16100i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.996431 + 1.16100i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.91 - 1.14i)T \) |
| 7 | \( 1 + (2.61 + 0.409i)T \) |
good | 3 | \( 1 + (-2.16 - 2.16i)T + 3iT^{2} \) |
| 11 | \( 1 - 0.796T + 11T^{2} \) |
| 13 | \( 1 + (-3.25 - 3.25i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.52 + 2.52i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.29T + 19T^{2} \) |
| 23 | \( 1 + (2.08 - 2.08i)T - 23iT^{2} \) |
| 29 | \( 1 + 10.0iT - 29T^{2} \) |
| 31 | \( 1 + 3.63iT - 31T^{2} \) |
| 37 | \( 1 + (-7.30 - 7.30i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.81iT - 41T^{2} \) |
| 43 | \( 1 + (-2.33 + 2.33i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.09 - 4.09i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.50 + 6.50i)T - 53iT^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 3.35iT - 61T^{2} \) |
| 67 | \( 1 + (2.49 + 2.49i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.93T + 71T^{2} \) |
| 73 | \( 1 + (-4.93 - 4.93i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.53iT - 79T^{2} \) |
| 83 | \( 1 + (-1.14 - 1.14i)T + 83iT^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + (-4.30 + 4.30i)T - 97iT^{2} \) |
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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
−11.86945723065735153496855680775, −11.08299833112579566756812754262, −9.838456268825641992296219634526, −9.542707623133014283601638734069, −8.362253099636227788311096185672, −7.53886388516474516260038410489, −6.21677439864828734020703336102, −4.39335349009538965446262952639, −3.68649044105491029658751603496, −2.80239403386180202442684453034,
1.14735913220121699054176173695, 3.00963776078436051367473370894, 3.75676985283322076168339288117, 5.81311004039665069099342873878, 6.96429597717396001548491572890, 7.80173546490119748870147966264, 8.596543872596022359425146545665, 9.261457409191515699774110648462, 10.66614218935954711327851284588, 12.13408966259213263140997216381