L(s) = 1 | + (−1.96 + 1.96i)2-s − 5.71i·4-s + (−1.65 − 1.50i)5-s + (0.707 + 0.707i)7-s + (7.29 + 7.29i)8-s + (6.20 − 0.280i)10-s − 0.248i·11-s + (−3.37 + 3.37i)13-s − 2.77·14-s − 17.2·16-s + (−1.75 + 1.75i)17-s + 3.91i·19-s + (−8.62 + 9.43i)20-s + (0.487 + 0.487i)22-s + (2.22 + 2.22i)23-s + ⋯ |
L(s) = 1 | + (−1.38 + 1.38i)2-s − 2.85i·4-s + (−0.738 − 0.674i)5-s + (0.267 + 0.267i)7-s + (2.58 + 2.58i)8-s + (1.96 − 0.0886i)10-s − 0.0749i·11-s + (−0.935 + 0.935i)13-s − 0.742·14-s − 4.31·16-s + (−0.426 + 0.426i)17-s + 0.898i·19-s + (−1.92 + 2.11i)20-s + (0.104 + 0.104i)22-s + (0.463 + 0.463i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00254683 + 0.304032i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00254683 + 0.304032i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.65 + 1.50i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (1.96 - 1.96i)T - 2iT^{2} \) |
| 11 | \( 1 + 0.248iT - 11T^{2} \) |
| 13 | \( 1 + (3.37 - 3.37i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.75 - 1.75i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.91iT - 19T^{2} \) |
| 23 | \( 1 + (-2.22 - 2.22i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.42iT - 41T^{2} \) |
| 43 | \( 1 + (0.716 - 0.716i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.34 - 3.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.25 + 4.25i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 - 9.55T + 61T^{2} \) |
| 67 | \( 1 + (5.98 + 5.98i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.31iT - 71T^{2} \) |
| 73 | \( 1 + (10.3 - 10.3i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.2iT - 79T^{2} \) |
| 83 | \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.91T + 89T^{2} \) |
| 97 | \( 1 + (-1.07 - 1.07i)T + 97iT^{2} \) |
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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
−11.79065782420169830566737291627, −10.98883766868273012105489278375, −9.756108135137313673374833329872, −9.121052601608403070625194111286, −8.233709167679507206646407781534, −7.57976406757558180243140238664, −6.61044841800991232646047011585, −5.44455737130909886230431871046, −4.49333194011255732932353609079, −1.61193533599050591218138447974,
0.34787960021115264521080163808, 2.37195428555062536177988178914, 3.33487525479844027178396481516, 4.58610190755550658910656627923, 7.06569230114367430353326842005, 7.55597450862528155359235186041, 8.550159685020705966320289144480, 9.486437575266456857741900190789, 10.48754710633574806629301643878, 10.97054261325874390452572482977