L(s) = 1 | + (−1.31 + 0.512i)2-s + (1.47 − 1.35i)4-s + (1.83 − 1.28i)5-s + (2.94 − 2.94i)7-s + (−1.25 + 2.53i)8-s + (−1.75 + 2.63i)10-s − 1.61·11-s + (−2.50 − 2.50i)13-s + (−2.37 + 5.39i)14-s + (0.350 − 3.98i)16-s + (−4.59 − 4.59i)17-s + 4i·19-s + (0.965 − 4.36i)20-s + (2.12 − 0.825i)22-s + (1.09 + 1.09i)23-s + ⋯ |
L(s) = 1 | + (−0.932 + 0.362i)2-s + (0.737 − 0.675i)4-s + (0.818 − 0.574i)5-s + (1.11 − 1.11i)7-s + (−0.442 + 0.896i)8-s + (−0.554 + 0.831i)10-s − 0.485·11-s + (−0.696 − 0.696i)13-s + (−0.635 + 1.44i)14-s + (0.0876 − 0.996i)16-s + (−1.11 − 1.11i)17-s + 0.917i·19-s + (0.215 − 0.976i)20-s + (0.452 − 0.176i)22-s + (0.227 + 0.227i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.939252 - 0.433038i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.939252 - 0.433038i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.31 - 0.512i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.83 + 1.28i)T \) |
good | 7 | \( 1 + (-2.94 + 2.94i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + (2.50 + 2.50i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.59 + 4.59i)T + 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (-1.09 - 1.09i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 - 5.01iT - 31T^{2} \) |
| 37 | \( 1 + (-2.50 + 2.50i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.18T + 41T^{2} \) |
| 43 | \( 1 + (7.40 - 7.40i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.32 + 7.32i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.11 + 3.11i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.61iT - 59T^{2} \) |
| 61 | \( 1 - 6.78iT - 61T^{2} \) |
| 67 | \( 1 + (-7.40 - 7.40i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 + (7.57 - 7.57i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.74iT - 89T^{2} \) |
| 97 | \( 1 + (-2.40 - 2.40i)T + 97iT^{2} \) |
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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
−10.94363458258321733093818117979, −10.31458465076782497847576029983, −9.504199334042061206375400999386, −8.438209663013682191938746537906, −7.69574681824276513845860209783, −6.78744537590954515923562865935, −5.43453624808188328945873972732, −4.68174665474264577252145418156, −2.40637245918648033406026770362, −0.996036944757701460424364819059,
1.97728256169692771069044397300, 2.60515337348381467003967154380, 4.61774521591427098425133202321, 5.96701277569136761881121479604, 6.91616703964353194604880420538, 8.072642922321876295455930886951, 8.907028408614205342958969302511, 9.625289244879719247951277340572, 10.78516196758074257398192066780, 11.22692850892670237180045153286