| L(s) = 1 | + (0.406 + 0.913i)2-s + (2.06 + 1.86i)3-s + (−0.669 + 0.743i)4-s + (0.677 − 2.13i)5-s + (−0.859 + 2.64i)6-s + (2.16 − 1.52i)7-s + (−0.951 − 0.309i)8-s + (0.495 + 4.71i)9-s + (2.22 − 0.247i)10-s + (−0.571 + 5.43i)11-s + (−2.76 + 0.290i)12-s + (−2.66 − 3.67i)13-s + (2.26 + 1.35i)14-s + (5.36 − 3.14i)15-s + (−0.104 − 0.994i)16-s + (−0.0537 − 0.252i)17-s + ⋯ |
| L(s) = 1 | + (0.287 + 0.645i)2-s + (1.19 + 1.07i)3-s + (−0.334 + 0.371i)4-s + (0.302 − 0.952i)5-s + (−0.350 + 1.08i)6-s + (0.818 − 0.574i)7-s + (−0.336 − 0.109i)8-s + (0.165 + 1.57i)9-s + (0.702 − 0.0783i)10-s + (−0.172 + 1.63i)11-s + (−0.798 + 0.0839i)12-s + (−0.739 − 1.01i)13-s + (0.606 + 0.363i)14-s + (1.38 − 0.811i)15-s + (−0.0261 − 0.248i)16-s + (−0.0130 − 0.0612i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73007 + 1.45779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73007 + 1.45779i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.406 - 0.913i)T \) |
| 5 | \( 1 + (-0.677 + 2.13i)T \) |
| 7 | \( 1 + (-2.16 + 1.52i)T \) |
| good | 3 | \( 1 + (-2.06 - 1.86i)T + (0.313 + 2.98i)T^{2} \) |
| 11 | \( 1 + (0.571 - 5.43i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (2.66 + 3.67i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.0537 + 0.252i)T + (-15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (1.04 + 1.15i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.62 - 3.65i)T + (-15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (2.92 + 9.01i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.44 - 1.37i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (0.913 - 0.0959i)T + (36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (2.95 - 2.14i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 6.07iT - 43T^{2} \) |
| 47 | \( 1 + (-1.91 + 9.02i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-3.16 - 2.85i)T + (5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-8.04 - 3.58i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-4.11 + 1.83i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-1.51 - 7.11i)T + (-61.2 + 27.2i)T^{2} \) |
| 71 | \( 1 + (0.884 + 2.72i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (14.1 + 1.48i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (1.20 + 0.256i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-12.9 - 4.20i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.3 + 5.03i)T + (59.5 - 66.1i)T^{2} \) |
| 97 | \( 1 + (-3.37 + 1.09i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.81839495122606577379544899806, −10.27451354290342492984754334769, −9.742098435322093216099511104123, −8.883482689758798699940585629346, −7.928353767164492862092222767229, −7.35953959716182895251138118635, −5.33639534323767796037489269583, −4.70249348405128140964933588006, −3.89767268321422849753832047394, −2.22078551371482142323334420965,
1.77189948891576902961098854042, 2.62286621082461438106621239334, 3.61069396417132452070206940729, 5.41607894493020298263537342061, 6.59717256934752242556289508669, 7.54660503720661789367061219663, 8.643324029495760388349445987968, 9.139135952895023311604610531611, 10.61505028745249978013068914916, 11.33156391550928113951598185033
