L(s) = 1 | + (0.622 − 1.26i)2-s + (−1.22 − 1.58i)4-s + 1.44·7-s + (−2.77 + 0.570i)8-s − 1.14i·11-s − 3.87i·13-s + (0.902 − 1.84i)14-s + (−0.999 + 3.87i)16-s + 3.05·17-s − 0.710i·19-s + (−1.44 − 0.710i)22-s − 5.54·23-s + (−4.91 − 2.41i)26-s + (−1.77 − 2.29i)28-s − 6.22i·29-s + ⋯ |
L(s) = 1 | + (0.440 − 0.897i)2-s + (−0.612 − 0.790i)4-s + 0.547·7-s + (−0.979 + 0.201i)8-s − 0.344i·11-s − 1.07i·13-s + (0.241 − 0.491i)14-s + (−0.249 + 0.968i)16-s + 0.739·17-s − 0.163i·19-s + (−0.309 − 0.151i)22-s − 1.15·23-s + (−0.964 − 0.472i)26-s + (−0.335 − 0.433i)28-s − 1.15i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.647543848\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.647543848\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.622 + 1.26i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 + 1.14iT - 11T^{2} \) |
| 13 | \( 1 + 3.87iT - 13T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 19 | \( 1 + 0.710iT - 19T^{2} \) |
| 23 | \( 1 + 5.54T + 23T^{2} \) |
| 29 | \( 1 + 6.22iT - 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 + 6.32iT - 37T^{2} \) |
| 41 | \( 1 + 8.03T + 41T^{2} \) |
| 43 | \( 1 + 7.03iT - 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 - 3.93iT - 53T^{2} \) |
| 59 | \( 1 - 10.1iT - 59T^{2} \) |
| 61 | \( 1 + 2.45iT - 61T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.980672757815059066629517563721, −8.235551459304228036194259079569, −7.50082845436964909498341010612, −6.10192584913329645488223187284, −5.59750334272438658918995521393, −4.67753061412772739943360685581, −3.77559467941451907212139011409, −2.88523934697075753531873328202, −1.82407933104550253481718032268, −0.53114621172528910950794458792,
1.62084915978282664017140808820, 3.04457038598820373208338532932, 4.09663975583774711447820856528, 4.78411171679165645298723478635, 5.60930243428872969696155120790, 6.53397470805062761938006166961, 7.14664434410359489696804944907, 8.084596172911586338008606538190, 8.532072779400093494348104910694, 9.554614156201027653917578738637