| L(s) = 1 | + (1.00 − 0.448i)2-s + (1.00 + 1.40i)3-s + (−0.524 + 0.582i)4-s + (1.72 − 1.41i)5-s + (1.64 + 0.966i)6-s + (0.860 − 1.49i)7-s + (−0.948 + 2.91i)8-s + (−0.965 + 2.84i)9-s + (1.10 − 2.20i)10-s + (0.0672 − 0.0299i)11-s + (−1.34 − 0.150i)12-s + (−2.55 − 1.13i)13-s + (0.198 − 1.88i)14-s + (3.74 + 1.00i)15-s + (0.190 + 1.80i)16-s + (0.268 − 0.827i)17-s + ⋯ |
| L(s) = 1 | + (0.712 − 0.317i)2-s + (0.582 + 0.812i)3-s + (−0.262 + 0.291i)4-s + (0.773 − 0.633i)5-s + (0.672 + 0.394i)6-s + (0.325 − 0.563i)7-s + (−0.335 + 1.03i)8-s + (−0.321 + 0.946i)9-s + (0.350 − 0.696i)10-s + (0.0202 − 0.00902i)11-s + (−0.389 − 0.0435i)12-s + (−0.709 − 0.315i)13-s + (0.0530 − 0.504i)14-s + (0.965 + 0.259i)15-s + (0.0475 + 0.452i)16-s + (0.0652 − 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98639 + 0.326927i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98639 + 0.326927i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.00 - 1.40i)T \) |
| 5 | \( 1 + (-1.72 + 1.41i)T \) |
| good | 2 | \( 1 + (-1.00 + 0.448i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (-0.860 + 1.49i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0672 + 0.0299i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (2.55 + 1.13i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-0.268 + 0.827i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.867 + 2.67i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.295 - 2.81i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (3.13 + 0.666i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (0.118 - 0.0251i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-1.86 + 1.35i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (10.1 + 4.49i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-4.98 + 8.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (10.7 + 2.29i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-2.96 - 9.13i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.95 - 3.09i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (11.6 - 5.19i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-15.7 + 3.34i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (-3.68 - 11.3i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.94 - 2.14i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.221 + 0.0470i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.50 - 2.78i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-1.26 - 0.916i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 2.15i)T + (88.6 + 39.4i)T^{2} \) |
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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
−12.49237842388069352855622664051, −11.40447780870857064638040253104, −10.30058796450583002426424766398, −9.379976211450070977945689198126, −8.567654558246783026997975698614, −7.48570283305107315594057612456, −5.47349530365869546384744280651, −4.81059633619758073642108461256, −3.73427935055750888754107429524, −2.36976148521657826018445418781,
1.90887339930004713370057555232, 3.33909273359371770107294899218, 5.02274632808050410692560134118, 6.12493451690788824418041194717, 6.82993521100066987566362284550, 8.115006245638423630286906756513, 9.333044818790711867989592715796, 10.01431702833094558080123130865, 11.50557292478606253628685480811, 12.57630181458765029508461893745
