| L(s) = 1 | + (−0.224 − 1.41i)2-s + (−1.54 − 0.786i)3-s + (1.84 − 0.599i)4-s + (−3.76 − 3.29i)5-s + (−0.768 + 2.36i)6-s + (−1.57 − 1.57i)7-s + (−3.87 − 7.59i)8-s + (1.76 + 2.42i)9-s + (−3.82 + 6.07i)10-s + (−2.65 − 1.92i)11-s + (−3.32 − 0.525i)12-s + (1.08 − 6.86i)13-s + (−1.87 + 2.58i)14-s + (3.21 + 8.03i)15-s + (−3.61 + 2.62i)16-s + (0.0769 − 0.0392i)17-s + ⋯ |
| L(s) = 1 | + (−0.112 − 0.708i)2-s + (−0.514 − 0.262i)3-s + (0.461 − 0.149i)4-s + (−0.752 − 0.658i)5-s + (−0.128 + 0.393i)6-s + (−0.224 − 0.224i)7-s + (−0.483 − 0.949i)8-s + (0.195 + 0.269i)9-s + (−0.382 + 0.607i)10-s + (−0.241 − 0.175i)11-s + (−0.276 − 0.0438i)12-s + (0.0836 − 0.528i)13-s + (−0.133 + 0.184i)14-s + (0.214 + 0.535i)15-s + (−0.225 + 0.164i)16-s + (0.00452 − 0.00230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.366112 - 0.874475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.366112 - 0.874475i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.54 + 0.786i)T \) |
| 5 | \( 1 + (3.76 + 3.29i)T \) |
| good | 2 | \( 1 + (0.224 + 1.41i)T + (-3.80 + 1.23i)T^{2} \) |
| 7 | \( 1 + (1.57 + 1.57i)T + 49iT^{2} \) |
| 11 | \( 1 + (2.65 + 1.92i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-1.08 + 6.86i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-0.0769 + 0.0392i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-30.3 - 9.85i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (-26.6 + 4.22i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-33.6 + 10.9i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-2.05 + 6.33i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (26.2 + 4.15i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (53.2 - 38.6i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (10.6 - 10.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (8.60 - 16.8i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-53.8 - 27.4i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (26.7 + 36.7i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (58.1 + 42.2i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-114. + 58.4i)T + (2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-23.5 - 72.4i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (87.6 - 13.8i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-136. + 44.4i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-66.3 - 130. i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (35.3 - 48.6i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (59.3 - 116. i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−13.49848953473882478866538271628, −12.44054072859054041960056747968, −11.74488305190809311511877081329, −10.78369892327618511259125870394, −9.660235878931774399015094013339, −8.036988757044694211501850490650, −6.78900347299368801692982366945, −5.20681645435729666375156049377, −3.31268601346317736582948077317, −0.950892496795529695299278095771,
3.13694700396151942390614622187, 5.15454080125101340180157476111, 6.66099614737436120620368439572, 7.38090394088862649651558489134, 8.824858518355978488550669830591, 10.42664168150371069884363209553, 11.50625361721300230302362040395, 12.14747431076814976039774039433, 13.90868446317908819842245366188, 15.14467140228814328694020065338
