L(s) = 1 | + (0.809 + 0.587i)2-s + (2.42 + 0.787i)3-s + (0.309 + 0.951i)4-s + (1.61 − 1.17i)5-s + (1.49 + 2.06i)6-s + (−2.37 + 0.770i)7-s + (−0.309 + 0.951i)8-s + (2.82 + 2.05i)9-s + 1.99·10-s + (−2.62 + 2.02i)11-s + 2.54i·12-s + (0.306 + 0.222i)13-s + (−2.37 − 0.770i)14-s + (4.83 − 1.57i)15-s + (−0.809 + 0.587i)16-s + (−1.98 − 2.72i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (1.39 + 0.454i)3-s + (0.154 + 0.475i)4-s + (0.722 − 0.525i)5-s + (0.611 + 0.841i)6-s + (−0.896 + 0.291i)7-s + (−0.109 + 0.336i)8-s + (0.941 + 0.684i)9-s + 0.631·10-s + (−0.792 + 0.610i)11-s + 0.735i·12-s + (0.0851 + 0.0618i)13-s + (−0.634 − 0.206i)14-s + (1.24 − 0.406i)15-s + (−0.202 + 0.146i)16-s + (−0.480 − 0.661i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.50428 + 1.28742i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.50428 + 1.28742i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (2.62 - 2.02i)T \) |
| 19 | \( 1 + (-1.14 + 4.20i)T \) |
good | 3 | \( 1 + (-2.42 - 0.787i)T + (2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-1.61 + 1.17i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (2.37 - 0.770i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.306 - 0.222i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.98 + 2.72i)T + (-5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 - 9.33T + 23T^{2} \) |
| 29 | \( 1 + (2.55 + 7.87i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.17 + 4.37i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (9.81 - 3.18i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.321 - 0.989i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 4.69iT - 43T^{2} \) |
| 47 | \( 1 + (3.57 - 11.0i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.76 + 5.18i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.80 - 1.56i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.57 - 3.53i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 4.35iT - 67T^{2} \) |
| 71 | \( 1 + (-0.523 - 0.720i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (11.9 - 3.88i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.93 - 7.21i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.00 - 4.14i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 2.43iT - 89T^{2} \) |
| 97 | \( 1 + (-7.02 + 9.66i)T + (-29.9 - 92.2i)T^{2} \) |
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
−11.41018410377522301766295008465, −10.02453185780216556867855289123, −9.326192097670347394679949706880, −8.823635576010412556900772559864, −7.65303697838426572138460618052, −6.71485228493883341972345337869, −5.39627980789773862947895173561, −4.51388342843301151171787747257, −3.13269759759446054164486408668, −2.40030073021782909445710306131,
1.78902488264240272270342479965, 3.04209614391620350423199609255, 3.44609877281370844525800996169, 5.26515734107012826736693938335, 6.45862100182439846980749795347, 7.22337884066211206020584405941, 8.471799726047605034335336080208, 9.228715728388503638348181653097, 10.31698766766288891250072124752, 10.77082496860148661069734750742