L(s) = 1 | + (−1.04 + 0.949i)2-s + (1.12 − 1.65i)3-s + (0.198 − 1.99i)4-s + (−0.117 + 1.56i)5-s + (0.388 + 2.80i)6-s + (1.24 + 2.33i)7-s + (1.68 + 2.27i)8-s + (−0.373 − 0.952i)9-s + (−1.36 − 1.74i)10-s + (−0.587 + 1.49i)11-s + (−3.07 − 2.57i)12-s + (2.03 + 2.55i)13-s + (−3.52 − 1.26i)14-s + (2.45 + 1.95i)15-s + (−3.92 − 0.789i)16-s + (−4.02 − 4.33i)17-s + ⋯ |
L(s) = 1 | + (−0.741 + 0.671i)2-s + (0.652 − 0.956i)3-s + (0.0991 − 0.995i)4-s + (−0.0523 + 0.698i)5-s + (0.158 + 1.14i)6-s + (0.471 + 0.881i)7-s + (0.594 + 0.804i)8-s + (−0.124 − 0.317i)9-s + (−0.430 − 0.553i)10-s + (−0.176 + 0.450i)11-s + (−0.887 − 0.743i)12-s + (0.564 + 0.707i)13-s + (−0.941 − 0.336i)14-s + (0.634 + 0.505i)15-s + (−0.980 − 0.197i)16-s + (−0.976 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11294 + 0.503636i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11294 + 0.503636i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.04 - 0.949i)T \) |
| 7 | \( 1 + (-1.24 - 2.33i)T \) |
good | 3 | \( 1 + (-1.12 + 1.65i)T + (-1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.117 - 1.56i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (0.587 - 1.49i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.03 - 2.55i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (4.02 + 4.33i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.91 - 2.26i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.12 - 1.21i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-2.66 + 0.607i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.55 - 2.69i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.01 + 6.51i)T + (-30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (3.96 + 8.22i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-0.651 - 0.313i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (1.86 - 0.280i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-2.26 - 7.32i)T + (-43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (2.41 - 0.181i)T + (58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (2.70 + 0.833i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-3.54 - 6.13i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.5 + 2.63i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.81 + 12.0i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (13.2 + 7.63i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.108 - 0.0868i)T + (18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-14.7 + 5.79i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 6.61iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.36557823176393329244779524201, −10.41729622811466263863945344875, −9.180959928667063130344165905338, −8.644080304744168342418013481100, −7.55574150781716661774615618850, −7.07235398968897357632361200285, −6.06969200275870926205068108842, −4.82205116731363837821129301216, −2.69577959493706666025494510189, −1.69530794436976896002472064557,
1.09586870157353784650386727905, 2.98626121525500567248123970366, 3.99812684514931813919360425466, 4.81266304900630471433067695429, 6.66323985700676869049810909995, 8.176338252885787426605262389443, 8.398769416219257086132876981669, 9.423582836401271008061853814811, 10.26220811337871584770777411350, 10.87365572146106539919842891633