L(s) = 1 | + (−1.41 + 0.0345i)2-s + (1.56 − 0.418i)3-s + (1.99 − 0.0977i)4-s + (−0.965 − 0.258i)5-s + (−2.19 + 0.645i)6-s + (−0.806 + 2.51i)7-s + (−2.82 + 0.207i)8-s + (−0.335 + 0.193i)9-s + (1.37 + 0.332i)10-s + (−0.0736 − 0.274i)11-s + (3.07 − 0.988i)12-s + (1.80 + 1.80i)13-s + (1.05 − 3.59i)14-s − 1.61·15-s + (3.98 − 0.390i)16-s + (−3.45 + 5.97i)17-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0244i)2-s + (0.901 − 0.241i)3-s + (0.998 − 0.0488i)4-s + (−0.431 − 0.115i)5-s + (−0.895 + 0.263i)6-s + (−0.304 + 0.952i)7-s + (−0.997 + 0.0732i)8-s + (−0.111 + 0.0645i)9-s + (0.434 + 0.105i)10-s + (−0.0221 − 0.0828i)11-s + (0.888 − 0.285i)12-s + (0.501 + 0.501i)13-s + (0.281 − 0.959i)14-s − 0.417·15-s + (0.995 − 0.0976i)16-s + (−0.837 + 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.689677 + 0.596549i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.689677 + 0.596549i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0345i)T \) |
| 5 | \( 1 + (0.965 + 0.258i)T \) |
| 7 | \( 1 + (0.806 - 2.51i)T \) |
good | 3 | \( 1 + (-1.56 + 0.418i)T + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (0.0736 + 0.274i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.80 - 1.80i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.45 - 5.97i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.90 - 7.12i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.59 + 2.07i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.292 + 0.292i)T + 29iT^{2} \) |
| 31 | \( 1 + (-4.94 + 8.55i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.46 - 2.26i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 3.42iT - 41T^{2} \) |
| 43 | \( 1 + (5.01 - 5.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.34 - 5.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.72 + 6.42i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (1.79 + 6.71i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (1.95 - 7.28i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (7.03 - 1.88i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 9.82iT - 71T^{2} \) |
| 73 | \( 1 + (-11.7 - 6.76i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.06 + 3.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.90 - 3.90i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.24 + 3.60i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−10.92555056637084310619953450762, −9.819342070537284161418608809911, −8.982411498590524926117629254801, −8.282259682721031580474851378205, −7.962497094677521934721214904851, −6.51229334808656839123238512314, −5.87531176442929985434699164914, −3.98356803257337219857492861157, −2.76966912699063220031619104398, −1.78526906437752036973193243782,
0.63142867658784617886512668193, 2.65989416493670900505769790900, 3.36708426005046757517854544889, 4.75349736072267862805853407763, 6.49921411769207583782690174074, 7.18295517808184116007936769615, 8.016949733194097043407305402956, 8.988789181741831562134206339752, 9.368126326177763810207191037968, 10.54487064234119710904640423843