| L(s) = 1 | + (0.514 + 1.31i)2-s + (0.617 − 0.882i)3-s + (−1.47 + 1.35i)4-s + (−0.208 + 2.38i)5-s + (1.47 + 0.359i)6-s + (1.04 − 1.81i)7-s + (−2.54 − 1.23i)8-s + (0.629 + 1.72i)9-s + (−3.24 + 0.950i)10-s + (−1.13 + 4.25i)11-s + (0.287 + 2.13i)12-s + (1.60 + 2.29i)13-s + (2.93 + 0.446i)14-s + (1.97 + 1.65i)15-s + (0.325 − 3.98i)16-s + (2.73 + 0.996i)17-s + ⋯ |
| L(s) = 1 | + (0.363 + 0.931i)2-s + (0.356 − 0.509i)3-s + (−0.735 + 0.677i)4-s + (−0.0931 + 1.06i)5-s + (0.604 + 0.146i)6-s + (0.396 − 0.686i)7-s + (−0.898 − 0.438i)8-s + (0.209 + 0.576i)9-s + (−1.02 + 0.300i)10-s + (−0.343 + 1.28i)11-s + (0.0829 + 0.616i)12-s + (0.445 + 0.636i)13-s + (0.783 + 0.119i)14-s + (0.508 + 0.427i)15-s + (0.0813 − 0.996i)16-s + (0.664 + 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.971359 + 1.26412i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.971359 + 1.26412i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.514 - 1.31i)T \) |
| 19 | \( 1 + (1.05 + 4.22i)T \) |
| good | 3 | \( 1 + (-0.617 + 0.882i)T + (-1.02 - 2.81i)T^{2} \) |
| 5 | \( 1 + (0.208 - 2.38i)T + (-4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-1.04 + 1.81i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.13 - 4.25i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.60 - 2.29i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-2.73 - 0.996i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (3.44 + 2.89i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.84 - 3.96i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-3.06 + 5.31i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.70 + 5.70i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.56 + 8.85i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.21 - 0.543i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-1.46 - 4.02i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (0.0422 + 0.483i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-5.63 + 2.62i)T + (37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (0.977 + 11.1i)T + (-60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (-8.86 - 4.13i)T + (43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (8.50 + 10.1i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (2.43 - 0.429i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.36 + 7.73i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.82 - 6.79i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.160 + 0.912i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-4.75 + 13.0i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−12.31368006168544733985780396695, −10.97979792288756923508211073082, −10.16621978938509731659210681684, −8.867215645885839947290722079660, −7.62596200299177460453546808890, −7.31434084507518381019507535306, −6.44187189024476817483165786411, −4.92778840316715061350053184444, −3.89545768397296531843793656656, −2.31717281230629008307312078862,
1.15435096313305567118772970636, 3.04018916607404297496991354630, 4.01248726582320940294057354424, 5.25304256516709903625560657590, 5.93814689904347862201316350888, 8.348923518381288978463730249221, 8.568445354801724833998552285485, 9.696281764827357120659836161517, 10.45910710427666018514714537858, 11.69948496603903384931595055877
