| L(s) = 1 | − 2.82i·2-s + 6.81i·3-s − 8.00·4-s + (6.15 + 24.2i)5-s + 19.2·6-s + 68.2·7-s + 22.6i·8-s + 34.5·9-s + (68.5 − 17.4i)10-s − 137. i·11-s − 54.5i·12-s + 37.8i·13-s − 193. i·14-s + (−165. + 41.9i)15-s + 64.0·16-s − 205.·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.757i·3-s − 0.500·4-s + (0.246 + 0.969i)5-s + 0.535·6-s + 1.39·7-s + 0.353i·8-s + 0.426·9-s + (0.685 − 0.174i)10-s − 1.13i·11-s − 0.378i·12-s + 0.223i·13-s − 0.984i·14-s + (−0.733 + 0.186i)15-s + 0.250·16-s − 0.709·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.302893615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.302893615\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 5 | \( 1 + (-6.15 - 24.2i)T \) |
| 23 | \( 1 + (-484. + 212. i)T \) |
| good | 3 | \( 1 - 6.81iT - 81T^{2} \) |
| 7 | \( 1 - 68.2T + 2.40e3T^{2} \) |
| 11 | \( 1 + 137. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 37.8iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 205.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 403. iT - 1.30e5T^{2} \) |
| 29 | \( 1 - 929.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.63e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.55e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.97e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 394.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.90e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.40e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 4.24e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 630. iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 492.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.92e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 6.58e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 739. iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 6.24e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 9.50e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 3.19e3T + 8.85e7T^{2} \) |
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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
−11.23138080723057749339100398003, −10.80659134471683429067846255784, −10.04813677888467276949452062937, −8.857605602597811183055975317554, −7.917823865172341670682622201197, −6.46038630734311360237392657926, −5.07525835316788164186673882622, −4.08788353603844972587335193274, −2.84883287899232626211413038288, −1.39004825267287790543866187680,
0.910087134700726464016252563666, 2.00445652915550642780158020209, 4.63300954489943232207192267651, 4.92407448507157330557299383252, 6.55386644959837273517651263477, 7.44791213963995731255855294365, 8.300290802836144728316859523669, 9.158715208556087801266421504036, 10.36864939618073928236516245019, 11.75477451166451627931511868334
