| L(s) = 1 | + (0.221 + 1.39i)2-s + (1.54 + 0.786i)3-s + (−1.90 + 0.618i)4-s + (4.68 − 1.74i)5-s + (−0.756 + 2.32i)6-s + (2.83 + 2.83i)7-s + (−1.28 − 2.52i)8-s + (1.76 + 2.42i)9-s + (3.47 + 6.15i)10-s + (6.47 + 4.70i)11-s + (−3.42 − 0.541i)12-s + (0.361 − 2.28i)13-s + (−3.32 + 4.57i)14-s + (8.60 + 0.984i)15-s + (3.23 − 2.35i)16-s + (−17.1 + 8.75i)17-s + ⋯ |
| L(s) = 1 | + (0.110 + 0.698i)2-s + (0.514 + 0.262i)3-s + (−0.475 + 0.154i)4-s + (0.936 − 0.349i)5-s + (−0.126 + 0.388i)6-s + (0.404 + 0.404i)7-s + (−0.160 − 0.315i)8-s + (0.195 + 0.269i)9-s + (0.347 + 0.615i)10-s + (0.588 + 0.427i)11-s + (−0.285 − 0.0451i)12-s + (0.0278 − 0.175i)13-s + (−0.237 + 0.327i)14-s + (0.573 + 0.0656i)15-s + (0.202 − 0.146i)16-s + (−1.01 + 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.63786 + 1.14147i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63786 + 1.14147i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.221 - 1.39i)T \) |
| 3 | \( 1 + (-1.54 - 0.786i)T \) |
| 5 | \( 1 + (-4.68 + 1.74i)T \) |
| good | 7 | \( 1 + (-2.83 - 2.83i)T + 49iT^{2} \) |
| 11 | \( 1 + (-6.47 - 4.70i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-0.361 + 2.28i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (17.1 - 8.75i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-3.83 - 1.24i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (8.54 - 1.35i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (32.8 - 10.6i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-14.4 + 44.5i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (24.9 + 3.95i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-28.4 + 20.6i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-46.8 + 46.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (13.6 - 26.7i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-6.49 - 3.30i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (57.3 + 78.9i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (32.9 + 23.9i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (71.4 - 36.4i)T + (2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (18.7 + 57.6i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-10.5 + 1.67i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-74.5 + 24.2i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-44.7 - 87.8i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (94.6 - 130. i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (49.9 - 98.0i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−13.23896354370583910051619275561, −12.26280018346691085567892003617, −10.78468956832493097691375805810, −9.499269748035447426700571154152, −8.937991547568728728047847961896, −7.79391606261445923490393975025, −6.42128990818139290420865668093, −5.33409771086205606490316468573, −4.09648941402836696060673332065, −2.06419077960215267680959138168,
1.54496392134653550176052306949, 2.93710695202949997175468368432, 4.46455559290275953050763986292, 6.03431735913240740847334570800, 7.25777798019836080139831808401, 8.742879088542937554952271375259, 9.510748081901422557677772875486, 10.63545063795410125529884997647, 11.48141009098944764374202978165, 12.70931340457938311847814216037
