| L(s) = 1 | + (−0.569 + 3.59i)3-s + (2.10 − 4.53i)5-s + (−0.635 − 0.635i)7-s + (−4.05 − 1.31i)9-s + (−4.24 − 13.0i)11-s + (−4.14 − 8.13i)13-s + (15.1 + 10.1i)15-s + (−0.920 − 5.81i)17-s + (18.5 − 25.4i)19-s + (2.64 − 1.92i)21-s + (−27.4 − 14.0i)23-s + (−16.1 − 19.1i)25-s + (−7.83 + 15.3i)27-s + (33.4 + 46.0i)29-s + (1.57 + 1.14i)31-s + ⋯ |
| L(s) = 1 | + (−0.189 + 1.19i)3-s + (0.421 − 0.906i)5-s + (−0.0908 − 0.0908i)7-s + (−0.450 − 0.146i)9-s + (−0.385 − 1.18i)11-s + (−0.318 − 0.625i)13-s + (1.00 + 0.677i)15-s + (−0.0541 − 0.341i)17-s + (0.974 − 1.34i)19-s + (0.126 − 0.0916i)21-s + (−1.19 − 0.608i)23-s + (−0.645 − 0.764i)25-s + (−0.290 + 0.569i)27-s + (1.15 + 1.58i)29-s + (0.0506 + 0.0368i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.30295 - 0.556202i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30295 - 0.556202i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.10 + 4.53i)T \) |
| good | 3 | \( 1 + (0.569 - 3.59i)T + (-8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (0.635 + 0.635i)T + 49iT^{2} \) |
| 11 | \( 1 + (4.24 + 13.0i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (4.14 + 8.13i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (0.920 + 5.81i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-18.5 + 25.4i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (27.4 + 14.0i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-33.4 - 46.0i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-1.57 - 1.14i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (10.4 - 5.33i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-20.6 + 63.6i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-38.8 + 38.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-49.2 - 7.80i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (5.69 - 35.9i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-57.8 - 18.7i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-4.37 - 13.4i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-10.1 - 63.8i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (25.4 - 18.4i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (61.7 + 31.4i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (33.4 + 45.9i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (40.0 - 6.33i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (57.1 - 18.5i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-12.3 - 1.94i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−10.63116966308318006194604546531, −10.17873512047319530478972065942, −9.088711112952848947587208023584, −8.589739393981140086143801010186, −7.22361440433634069379898106479, −5.67107883205802192639665436534, −5.15593746314607488437166825592, −4.11231189726833256004965242803, −2.81422411142273255890808037445, −0.64535011853794903059951187589,
1.60790471578137838649278247592, 2.54373408148734028576926590680, 4.19027888389503883943643688236, 5.81294886800118688951342242493, 6.47241515923799993191383449969, 7.46944414199074598632134698643, 7.925257063542711553909766719736, 9.762999191073231759418276004837, 9.995435059606697041831892513762, 11.43128526798166361941751064537
