| L(s) = 1 | + (0.368 − 0.637i)2-s + (1.41 − 0.516i)3-s + (0.729 + 1.26i)4-s + (−2.11 − 1.77i)5-s + (0.192 − 1.09i)6-s + (1.03 + 0.866i)7-s + 2.54·8-s + (−0.553 + 0.464i)9-s + (−1.90 + 0.693i)10-s + (−4.58 − 1.66i)11-s + (1.68 + 1.41i)12-s + (1.09 + 0.921i)13-s + (0.932 − 0.339i)14-s + (−3.90 − 1.42i)15-s + (−0.521 + 0.903i)16-s + (−0.990 + 1.71i)17-s + ⋯ |
| L(s) = 1 | + (0.260 − 0.450i)2-s + (0.818 − 0.298i)3-s + (0.364 + 0.631i)4-s + (−0.944 − 0.792i)5-s + (0.0787 − 0.446i)6-s + (0.390 + 0.327i)7-s + 0.899·8-s + (−0.184 + 0.154i)9-s + (−0.602 + 0.219i)10-s + (−1.38 − 0.503i)11-s + (0.486 + 0.408i)12-s + (0.304 + 0.255i)13-s + (0.249 − 0.0907i)14-s + (−1.00 − 0.367i)15-s + (−0.130 + 0.225i)16-s + (−0.240 + 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31340 - 0.392702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31340 - 0.392702i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (10.4 + 0.871i)T \) |
| good | 2 | \( 1 + (-0.368 + 0.637i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.41 + 0.516i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (2.11 + 1.77i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.03 - 0.866i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (4.58 + 1.66i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.09 - 0.921i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.990 - 1.71i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.56 + 2.71i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.255 - 0.443i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.43 - 1.25i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.24 - 2.72i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 1.09i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 + (-4.52 + 7.83i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.237 + 1.34i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (9.30 + 7.80i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-13.2 - 4.82i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.59 + 9.02i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.80 + 2.35i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.22 - 5.58i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.12 + 2.95i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (11.7 - 9.86i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.94 + 1.07i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.00 + 11.3i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-13.4 - 11.2i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.22325164016078732762019027241, −12.76838157041740708170902335921, −11.54898825626768242108283285996, −10.87184543463838254172962574995, −8.842204304647111867928031010701, −8.152151599347810700043368066925, −7.43317912139426534699009659906, −5.18804551079175785705051756581, −3.71972323594654417200917248112, −2.37650200782763530187921596840,
2.74511855390920647001336933606, 4.28347178596524700746815057058, 5.82739697407228823141731704141, 7.43664389756573287664008823994, 7.88033001939883287133374862112, 9.590446768264954868130413298221, 10.71752058624116862821199023166, 11.40140775721440493709234159606, 13.04609987404793143654930748705, 14.27423554650788931545956096421
