| L(s) = 1 | + (−1.39 − 0.221i)2-s + (−1.36 − 2.68i)3-s + (1.90 + 0.618i)4-s + (1.31 + 4.04i)6-s + (−3.67 − 3.67i)7-s + (−2.52 − 1.28i)8-s + (−0.0358 + 0.0492i)9-s + (−4.26 + 3.09i)11-s + (−0.941 − 5.94i)12-s + (−9.64 + 1.52i)13-s + (4.32 + 5.94i)14-s + (3.23 + 2.35i)16-s + (10.0 − 19.6i)17-s + (0.0609 − 0.0609i)18-s + (−11.5 + 3.74i)19-s + ⋯ |
| L(s) = 1 | + (−0.698 − 0.110i)2-s + (−0.455 − 0.894i)3-s + (0.475 + 0.154i)4-s + (0.219 + 0.674i)6-s + (−0.524 − 0.524i)7-s + (−0.315 − 0.160i)8-s + (−0.00397 + 0.00547i)9-s + (−0.387 + 0.281i)11-s + (−0.0784 − 0.495i)12-s + (−0.742 + 0.117i)13-s + (0.308 + 0.424i)14-s + (0.202 + 0.146i)16-s + (0.589 − 1.15i)17-s + (0.00338 − 0.00338i)18-s + (−0.606 + 0.197i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0485647 + 0.115207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0485647 + 0.115207i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.221i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (1.36 + 2.68i)T + (-5.29 + 7.28i)T^{2} \) |
| 7 | \( 1 + (3.67 + 3.67i)T + 49iT^{2} \) |
| 11 | \( 1 + (4.26 - 3.09i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (9.64 - 1.52i)T + (160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-10.0 + 19.6i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (11.5 - 3.74i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (6.26 - 39.5i)T + (-503. - 163. i)T^{2} \) |
| 29 | \( 1 + (29.8 + 9.71i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-14.9 - 45.9i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-9.11 - 57.5i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (32.9 + 23.9i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (31.9 - 31.9i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (19.2 - 9.82i)T + (1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (34.4 + 67.5i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-16.7 + 23.0i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-41.1 + 29.8i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (5.59 - 10.9i)T + (-2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-30.7 + 94.5i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-0.568 + 3.58i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (76.6 + 24.8i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-63.7 - 32.5i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (32.5 + 44.7i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-88.4 + 45.0i)T + (5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.41472328235477835159336376990, −10.02683570512435034523850112888, −9.596704445693062184045921196826, −8.049837621537617903197389768682, −7.21285634907118055922464194754, −6.57378517798055171474595419791, −5.17788189209993232392269838233, −3.32653029225247960434959183087, −1.63620354269576383252931517686, −0.083384153452994780000861016094,
2.39717002406506056601101188744, 4.06616638880897816514505543623, 5.40917245138393880435725987237, 6.30288744658980300356940021198, 7.68605969020747748696980258538, 8.694496932582904539949741401059, 9.739463180205649169033195813624, 10.37673053422437350583823321410, 11.11864672592726234737532250676, 12.30754583432977721393596800491
