| L(s) = 1 | + (−1.32 − 0.481i)2-s + (0.327 + 1.00i)3-s + (1.53 + 1.27i)4-s + (1.12 − 1.92i)5-s + (0.0494 − 1.49i)6-s + 3.62i·7-s + (−1.42 − 2.44i)8-s + (1.51 − 1.10i)9-s + (−2.43 + 2.02i)10-s + (−0.116 + 0.160i)11-s + (−0.786 + 1.96i)12-s + (−1.57 + 1.14i)13-s + (1.74 − 4.81i)14-s + (2.31 + 0.506i)15-s + (0.724 + 3.93i)16-s + (5.99 + 1.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.940 − 0.340i)2-s + (0.189 + 0.582i)3-s + (0.768 + 0.639i)4-s + (0.505 − 0.863i)5-s + (0.0201 − 0.611i)6-s + 1.36i·7-s + (−0.504 − 0.863i)8-s + (0.506 − 0.367i)9-s + (−0.768 + 0.639i)10-s + (−0.0351 + 0.0483i)11-s + (−0.227 + 0.568i)12-s + (−0.437 + 0.318i)13-s + (0.465 − 1.28i)14-s + (0.597 + 0.130i)15-s + (0.181 + 0.983i)16-s + (1.45 + 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.943648 + 0.136458i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.943648 + 0.136458i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 + 0.481i)T \) |
| 5 | \( 1 + (-1.12 + 1.92i)T \) |
| good | 3 | \( 1 + (-0.327 - 1.00i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 3.62iT - 7T^{2} \) |
| 11 | \( 1 + (0.116 - 0.160i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.57 - 1.14i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.99 - 1.94i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-3.31 - 1.07i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.186 - 0.256i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.98 - 0.644i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.40 + 7.39i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (3.14 - 2.28i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.48 - 1.80i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + (7.86 - 2.55i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (4.48 + 13.7i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (0.165 + 0.227i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (1.86 - 2.57i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.0384 - 0.118i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.42 + 4.38i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.51 + 10.3i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.42 - 4.37i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.15 + 12.7i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (13.8 + 10.0i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.24 - 0.730i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.23279882492246734972297966267, −11.67442816326009318277736573047, −9.929897836364896219387779684536, −9.730193251610508003109898937978, −8.773059685505468046962100193522, −7.88269544345139597578817114096, −6.27661896425120852444519005570, −5.07566391696818561627587384522, −3.36207645041548694055793887593, −1.73202913224184551182793631306,
1.38062260975997166518977971425, 3.08683796067398662873878906092, 5.27313389808743626037578571208, 6.79884085325008438952392278890, 7.27730943186584785635059728311, 8.063065060237393601458374291798, 9.775808483454085860512731555118, 10.19545514941852871648090368200, 11.07789696595014264475101833348, 12.34426636209692980659023919500
