| L(s) = 1 | + (0.366 + 0.719i)3-s + (−0.581 − 2.15i)5-s + (1.82 + 1.82i)7-s + (1.37 − 1.89i)9-s + (−2.56 − 3.52i)11-s + (−0.435 − 2.74i)13-s + (1.34 − 1.21i)15-s + (−3.87 − 1.97i)17-s + (−0.914 − 2.81i)19-s + (−0.642 + 1.97i)21-s + (−0.658 + 4.15i)23-s + (−4.32 + 2.51i)25-s + (4.26 + 0.675i)27-s + (1.54 + 0.503i)29-s + (10.0 − 3.25i)31-s + ⋯ |
| L(s) = 1 | + (0.211 + 0.415i)3-s + (−0.260 − 0.965i)5-s + (0.688 + 0.688i)7-s + (0.459 − 0.633i)9-s + (−0.772 − 1.06i)11-s + (−0.120 − 0.762i)13-s + (0.346 − 0.312i)15-s + (−0.940 − 0.479i)17-s + (−0.209 − 0.645i)19-s + (−0.140 + 0.431i)21-s + (−0.137 + 0.866i)23-s + (−0.864 + 0.502i)25-s + (0.821 + 0.130i)27-s + (0.287 + 0.0934i)29-s + (1.79 − 0.583i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20190 - 0.851603i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20190 - 0.851603i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.581 + 2.15i)T \) |
| good | 3 | \( 1 + (-0.366 - 0.719i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-1.82 - 1.82i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.56 + 3.52i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.435 + 2.74i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (3.87 + 1.97i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.914 + 2.81i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.658 - 4.15i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.54 - 0.503i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-10.0 + 3.25i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.29 + 0.521i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (8.06 + 5.86i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.01 + 2.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.681 - 0.347i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (4.65 - 2.37i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (0.111 + 0.0810i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.77 + 7.10i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.64 + 5.19i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (0.559 + 0.181i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-15.0 - 2.38i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (3.31 - 10.1i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.0483 - 0.0246i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-8.88 - 12.2i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.19 + 10.1i)T + (-57.0 + 78.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.957026346449480323423639903259, −9.140658471616190965786231161233, −8.437653247536228011419079026986, −7.896947671611974467857865070937, −6.53678243877993749032021096670, −5.37136402367076535630046124242, −4.82574623824258095186532461805, −3.68829667992421747797019336709, −2.46499658894099140313498655818, −0.72819696122503841075720534881,
1.75902572221140903975564417175, 2.64205415895498533742796837960, 4.26403450524420778163202299313, 4.73554558967451093186345424433, 6.46397098995040338683171626477, 6.98573712179029866807075097418, 7.87186867996742881365570982009, 8.346820739928880187188396258234, 9.962136721786674688120892214011, 10.36377496682543410823898965911
