L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (2.09 + 3.62i)5-s + (−1.62 − 2.09i)7-s − 0.999i·8-s + (3.62 + 2.09i)10-s + (−2.59 − 1.5i)11-s − 2.44i·13-s + (−2.44 − 0.999i)14-s + (−0.5 − 0.866i)16-s + (−0.507 + 0.878i)17-s + (−0.878 + 0.507i)19-s + 4.18·20-s − 3·22-s + (−3.67 + 2.12i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.935 + 1.61i)5-s + (−0.612 − 0.790i)7-s − 0.353i·8-s + (1.14 + 0.661i)10-s + (−0.783 − 0.452i)11-s − 0.679i·13-s + (−0.654 − 0.267i)14-s + (−0.125 − 0.216i)16-s + (−0.123 + 0.213i)17-s + (−0.201 + 0.116i)19-s + 0.935·20-s − 0.639·22-s + (−0.766 + 0.442i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49602 - 0.126689i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49602 - 0.126689i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.62 + 2.09i)T \) |
good | 5 | \( 1 + (-2.09 - 3.62i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (0.507 - 0.878i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.878 - 0.507i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.24iT - 29T^{2} \) |
| 31 | \( 1 + (-4.86 - 2.80i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.12 + 7.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 + (0.507 + 0.878i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.07 - 0.621i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.76 - 9.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.12 + 2.95i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 + (-7.24 - 4.18i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.62 - 9.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.76iT - 97T^{2} \) |
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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
−13.65014528747896657549649463703, −12.53573651810426380156954457905, −10.92055016454385686452644191908, −10.52820495678704976121793752481, −9.676187988778515871465835758364, −7.63394240889213259180219884345, −6.52254269220323586442557272963, −5.65278745410887970585731553620, −3.63075997645600839621479756175, −2.53910575083662099604526767520,
2.28204504192451370250960903204, 4.50966455895037023794726880306, 5.45091027992580206218923318806, 6.44279159329338704180310005479, 8.187042968845198272548302864760, 9.123919097793103516996065733477, 10.02760840387882736476277039588, 11.86885112365780663082114627898, 12.63617714891846620471241645384, 13.25292213632619626975872722678