| L(s) = 1 | + (−4.57 + 3.32i)3-s + (−3.76 − 11.5i)5-s + (27.3 + 19.8i)7-s + (1.51 − 4.67i)9-s + (−34.2 − 12.4i)11-s + (−6.38 + 19.6i)13-s + (55.6 + 40.4i)15-s + (−18.5 − 57.2i)17-s + (−85.6 + 62.2i)19-s − 191.·21-s − 186.·23-s + (−18.7 + 13.6i)25-s + (−38.5 − 118. i)27-s + (−109. − 79.5i)29-s + (−27.6 + 85.2i)31-s + ⋯ |
| L(s) = 1 | + (−0.879 + 0.639i)3-s + (−0.336 − 1.03i)5-s + (1.47 + 1.07i)7-s + (0.0562 − 0.173i)9-s + (−0.939 − 0.342i)11-s + (−0.136 + 0.419i)13-s + (0.957 + 0.695i)15-s + (−0.265 − 0.816i)17-s + (−1.03 + 0.751i)19-s − 1.98·21-s − 1.69·23-s + (−0.150 + 0.109i)25-s + (−0.274 − 0.845i)27-s + (−0.701 − 0.509i)29-s + (−0.160 + 0.493i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0106i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00132232 - 0.249186i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00132232 - 0.249186i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (34.2 + 12.4i)T \) |
| good | 3 | \( 1 + (4.57 - 3.32i)T + (8.34 - 25.6i)T^{2} \) |
| 5 | \( 1 + (3.76 + 11.5i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-27.3 - 19.8i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (6.38 - 19.6i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (18.5 + 57.2i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (85.6 - 62.2i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 186.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (109. + 79.5i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (27.6 - 85.2i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-148. - 108. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (301. - 219. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-6.86 + 4.98i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (67.9 - 209. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-210. - 153. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (159. + 489. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 263.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-95.6 - 294. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-170. - 123. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-288. + 888. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-73.5 - 226. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 620.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-372. + 1.14e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−12.34145287782639887001010733077, −11.71395572091987843306763270432, −10.99981574504129431537296045565, −9.832865660200632955679737564190, −8.459994307029439114646982497483, −8.042335243601932589139999082792, −5.92881125737313697680724689710, −5.06227873909654785406083896311, −4.45549828142160470748230147866, −2.01790990768231636747848030437,
0.12000654947139637312823181128, 1.95507604698753929826131991037, 3.96329575176438314996573478749, 5.29426997307019655591971296924, 6.61681852618210349103389064043, 7.45343275366247768634534970820, 8.231667631899586653630621681906, 10.30546499645863582723192352250, 10.88561996153039071849588366882, 11.50550498997867255725476079560
