L(s) = 1 | + (−1.30 − 0.665i)2-s + (−0.822 + 0.130i)3-s + (0.0875 + 0.120i)4-s + (−0.233 + 2.22i)5-s + (1.16 + 0.377i)6-s + (−0.659 + 4.16i)7-s + (0.424 + 2.68i)8-s + (−2.19 + 0.712i)9-s + (1.78 − 2.74i)10-s + (−0.0877 − 0.0877i)12-s + (0.420 − 0.824i)13-s + (3.63 − 4.99i)14-s + (−0.0979 − 1.85i)15-s + (1.32 − 4.06i)16-s + (−0.875 − 1.71i)17-s + (3.34 + 0.529i)18-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.470i)2-s + (−0.474 + 0.0751i)3-s + (0.0437 + 0.0602i)4-s + (−0.104 + 0.994i)5-s + (0.473 + 0.153i)6-s + (−0.249 + 1.57i)7-s + (0.150 + 0.947i)8-s + (−0.731 + 0.237i)9-s + (0.564 − 0.869i)10-s + (−0.0253 − 0.0253i)12-s + (0.116 − 0.228i)13-s + (0.970 − 1.33i)14-s + (−0.0252 − 0.479i)15-s + (0.330 − 1.01i)16-s + (−0.212 − 0.416i)17-s + (0.787 + 0.124i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00700310 - 0.0342484i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00700310 - 0.0342484i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.233 - 2.22i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.30 + 0.665i)T + (1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (0.822 - 0.130i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (0.659 - 4.16i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (-0.420 + 0.824i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.875 + 1.71i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (4.39 + 3.19i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.95 + 1.95i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.810 + 0.588i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.131 - 0.403i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.87 + 0.771i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (0.339 - 0.467i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.05 - 5.05i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.186 + 1.17i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-8.09 - 4.12i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (5.47 + 7.53i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (7.40 + 2.40i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (3.05 + 3.05i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.65 + 8.17i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (5.39 + 0.854i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (0.705 + 2.17i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.77 - 0.902i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 - 13.9iT - 89T^{2} \) |
| 97 | \( 1 + (-1.50 + 2.96i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−10.95220557214777442068399562593, −10.47541999872515543403532021887, −9.318720680977620113706905467743, −8.794036988969581409908061383468, −7.924032110495926937079591734087, −6.55047854403354174551373879900, −5.80069082954911437016539237825, −4.87588684597371856060416235786, −2.90824596945471983471592041389, −2.27255252967304630796326189170,
0.03102957122990552734210532822, 1.21196350895557407146672307483, 3.68708179576497800988258065918, 4.41906627034415423408468530744, 5.80723874519734828444067508103, 6.77877808360679536456749025542, 7.57756376812068018483438928449, 8.498026914755046185423726464231, 9.043680363590678571086242293644, 10.14382451950367151178602985162