| L(s) = 1 | + (−1.39 − 0.221i)2-s + (−0.786 − 1.54i)3-s + (1.90 + 0.618i)4-s + (4.99 + 0.204i)5-s + (0.756 + 2.32i)6-s + (1.89 + 1.89i)7-s + (−2.52 − 1.28i)8-s + (−1.76 + 2.42i)9-s + (−6.93 − 1.39i)10-s + (−6.63 + 4.82i)11-s + (−0.541 − 3.42i)12-s + (22.2 − 3.53i)13-s + (−2.22 − 3.06i)14-s + (−3.61 − 7.87i)15-s + (3.23 + 2.35i)16-s + (10.4 − 20.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 − 0.110i)2-s + (−0.262 − 0.514i)3-s + (0.475 + 0.154i)4-s + (0.999 + 0.0409i)5-s + (0.126 + 0.388i)6-s + (0.270 + 0.270i)7-s + (−0.315 − 0.160i)8-s + (−0.195 + 0.269i)9-s + (−0.693 − 0.139i)10-s + (−0.603 + 0.438i)11-s + (−0.0451 − 0.285i)12-s + (1.71 − 0.271i)13-s + (−0.158 − 0.218i)14-s + (−0.240 − 0.524i)15-s + (0.202 + 0.146i)16-s + (0.613 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15172 - 0.346516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15172 - 0.346516i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.221i)T \) |
| 3 | \( 1 + (0.786 + 1.54i)T \) |
| 5 | \( 1 + (-4.99 - 0.204i)T \) |
| good | 7 | \( 1 + (-1.89 - 1.89i)T + 49iT^{2} \) |
| 11 | \( 1 + (6.63 - 4.82i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-22.2 + 3.53i)T + (160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-10.4 + 20.4i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-13.3 + 4.32i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-3.43 + 21.6i)T + (-503. - 163. i)T^{2} \) |
| 29 | \( 1 + (41.3 + 13.4i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-14.3 - 44.0i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (3.81 + 24.0i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-45.0 - 32.7i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (17.8 - 17.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (56.1 - 28.6i)T + (1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-13.3 - 26.1i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (23.5 - 32.3i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (36.2 - 26.3i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-27.4 + 53.8i)T + (-2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (5.74 - 17.6i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (16.0 - 101. i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (100. + 32.7i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-33.1 - 16.8i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (0.221 + 0.304i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (36.2 - 18.4i)T + (5.53e3 - 7.61e3i)T^{2} \) |
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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
−12.71395903518325478219091991771, −11.49651564128205475571157114935, −10.65655018093334717231142010926, −9.600567763484815682861174068973, −8.593013188518488880224820459531, −7.44939717641936663270122690211, −6.26564040147037276653841431658, −5.24208868371479110242821036884, −2.81222099050810458045980262671, −1.28066322728037662194326508523,
1.47905183471457606483280893912, 3.57191039918062630332330232764, 5.50705105322482520826028867007, 6.19484350136781866962246321266, 7.81531795323631943843080405411, 8.877561380391632618552320281048, 9.837972332852883941738028022587, 10.72771227731121660993856584056, 11.41332670282599459793532556640, 13.01534863804717265906538189683
