L(s) = 1 | + (1.32 − 2.29i)3-s + (1.5 − 0.866i)5-s − 2.64·7-s + (−2 − 3.46i)9-s + (−3.96 − 2.29i)11-s − 3.46i·13-s − 4.58i·15-s + (4.5 + 2.59i)17-s + (1.32 + 2.29i)19-s + (−3.50 + 6.06i)21-s + (3.96 − 2.29i)23-s + (−1 + 1.73i)25-s − 2.64·27-s + (1.32 − 2.29i)31-s + (−10.5 + 6.06i)33-s + ⋯ |
L(s) = 1 | + (0.763 − 1.32i)3-s + (0.670 − 0.387i)5-s − 0.999·7-s + (−0.666 − 1.15i)9-s + (−1.19 − 0.690i)11-s − 0.960i·13-s − 1.18i·15-s + (1.09 + 0.630i)17-s + (0.303 + 0.525i)19-s + (−0.763 + 1.32i)21-s + (0.827 − 0.477i)23-s + (−0.200 + 0.346i)25-s − 0.509·27-s + (0.237 − 0.411i)31-s + (−1.82 + 1.05i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.926713 - 1.39317i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.926713 - 1.39317i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 3 | \( 1 + (-1.32 + 2.29i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.96 + 2.29i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-4.5 - 2.59i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-1.32 + 2.29i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 9.16iT - 43T^{2} \) |
| 47 | \( 1 + (-3.96 - 6.87i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 0.866i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.96 - 2.29i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.16iT - 71T^{2} \) |
| 73 | \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.96 + 2.29i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61228835978616700584541547220, −9.888013900395411405773847271451, −8.796538602939488660827700703048, −8.025061254487974546172989366329, −7.30190410586901942301832239600, −6.07847175169616642711416352293, −5.47556899946297274459248552212, −3.34520832783832062512994153738, −2.55504560494110432051271872280, −0.999125218425913087783433789104,
2.51876891507644918163544429969, 3.27061291014582210909593007890, 4.54157641700522856174806261109, 5.49045783669412762414320363526, 6.76822808675233184757884014788, 7.79863633494701054888935615476, 9.151523105135185430978033530806, 9.624301997540606079359239143350, 10.15054397986786421086910394979, 11.00663386241407087987032415680