L(s) = 1 | + i·2-s − 4-s − 0.260·5-s + (−1.84 − 1.89i)7-s − i·8-s − 0.260i·10-s + i·11-s + 5.36i·13-s + (1.89 − 1.84i)14-s + 16-s + 4.05·17-s − 0.629i·19-s + 0.260·20-s − 22-s − 4.93·25-s − 5.36·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.116·5-s + (−0.697 − 0.716i)7-s − 0.353i·8-s − 0.0824i·10-s + 0.301i·11-s + 1.48i·13-s + (0.506 − 0.493i)14-s + 0.250·16-s + 0.983·17-s − 0.144i·19-s + 0.0583·20-s − 0.213·22-s − 0.986·25-s − 1.05·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4657914314\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4657914314\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.84 + 1.89i)T \) |
| 11 | \( 1 - iT \) |
good | 5 | \( 1 + 0.260T + 5T^{2} \) |
| 13 | \( 1 - 5.36iT - 13T^{2} \) |
| 17 | \( 1 - 4.05T + 17T^{2} \) |
| 19 | \( 1 + 0.629iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4.94iT - 29T^{2} \) |
| 31 | \( 1 - 4.42iT - 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 + 9.78T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 - 5.30iT - 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 2.94T + 67T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 5.62T + 79T^{2} \) |
| 83 | \( 1 + 0.0507T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 - 0.737iT - 97T^{2} \) |
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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
−9.876562913611432453386669437202, −9.207393533086708639381887482930, −8.287304747844810036113185181265, −7.40352746581361302043761591105, −6.79834195529964964926509674892, −6.09768344542885597602668929272, −4.96945505781268402091996217851, −4.11226959368092840071244387773, −3.27776553302945874920291613349, −1.59932473615433069307877982831,
0.18773704820667988327233071975, 1.79039510803660460522198372216, 3.19790228198835029383332263595, 3.43687718886876346196083377611, 5.04486356246016857339750198894, 5.63299073680617800974312158717, 6.57074849772036751562172671227, 7.83923414488901309670933961679, 8.362603232385518121951483974354, 9.330852853202155885915383961090