| L(s) = 1 | + (−0.203 − 0.140i)2-s + (0.885 − 0.464i)3-s + (−0.687 − 1.81i)4-s + (2.06 − 2.32i)5-s + (−0.245 − 0.0298i)6-s + (−0.296 + 1.20i)7-s + (−0.233 + 0.947i)8-s + (0.568 − 0.822i)9-s + (−0.748 + 0.184i)10-s + (1.56 − 1.07i)11-s + (−1.45 − 1.28i)12-s + (3.56 + 0.548i)13-s + (0.229 − 0.203i)14-s + (0.744 − 3.02i)15-s + (−2.72 + 2.41i)16-s + (0.475 + 0.117i)17-s + ⋯ |
| L(s) = 1 | + (−0.144 − 0.0994i)2-s + (0.511 − 0.268i)3-s + (−0.343 − 0.906i)4-s + (0.923 − 1.04i)5-s + (−0.100 − 0.0121i)6-s + (−0.111 + 0.454i)7-s + (−0.0825 + 0.334i)8-s + (0.189 − 0.274i)9-s + (−0.236 + 0.0583i)10-s + (0.470 − 0.324i)11-s + (−0.418 − 0.371i)12-s + (0.988 + 0.152i)13-s + (0.0613 − 0.0543i)14-s + (0.192 − 0.780i)15-s + (−0.680 + 0.602i)16-s + (0.115 + 0.0284i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0729 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0729 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15081 - 1.23804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15081 - 1.23804i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.885 + 0.464i)T \) |
| 13 | \( 1 + (-3.56 - 0.548i)T \) |
| good | 2 | \( 1 + (0.203 + 0.140i)T + (0.709 + 1.87i)T^{2} \) |
| 5 | \( 1 + (-2.06 + 2.32i)T + (-0.602 - 4.96i)T^{2} \) |
| 7 | \( 1 + (0.296 - 1.20i)T + (-6.19 - 3.25i)T^{2} \) |
| 11 | \( 1 + (-1.56 + 1.07i)T + (3.90 - 10.2i)T^{2} \) |
| 17 | \( 1 + (-0.475 - 0.117i)T + (15.0 + 7.90i)T^{2} \) |
| 19 | \( 1 + 3.70iT - 19T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 + (2.93 - 4.25i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (6.12 + 0.744i)T + (30.0 + 7.41i)T^{2} \) |
| 37 | \( 1 + (8.87 + 1.07i)T + (35.9 + 8.85i)T^{2} \) |
| 41 | \( 1 + (-4.60 - 8.78i)T + (-23.2 + 33.7i)T^{2} \) |
| 43 | \( 1 + (1.17 + 9.66i)T + (-41.7 + 10.2i)T^{2} \) |
| 47 | \( 1 + (-0.298 - 0.113i)T + (35.1 + 31.1i)T^{2} \) |
| 53 | \( 1 + (-9.72 - 2.39i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (-3.02 + 3.41i)T + (-7.11 - 58.5i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 0.378i)T + (54.0 - 28.3i)T^{2} \) |
| 67 | \( 1 + (-9.43 - 3.57i)T + (50.1 + 44.4i)T^{2} \) |
| 71 | \( 1 + (-5.50 - 10.4i)T + (-40.3 + 58.4i)T^{2} \) |
| 73 | \( 1 + (9.37 - 6.47i)T + (25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-3.28 + 8.65i)T + (-59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (-2.36 + 4.50i)T + (-47.1 - 68.3i)T^{2} \) |
| 89 | \( 1 - 16.3iT - 89T^{2} \) |
| 97 | \( 1 + (5.49 + 6.20i)T + (-11.6 + 96.2i)T^{2} \) |
| 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
−10.51207682724171330356439992847, −9.496136826125984504910286054665, −8.950250913890681856476966131632, −8.522538757336711686716000050689, −6.87626009812350759902893012137, −5.82082361603756879423241852886, −5.25967305216425253606474003847, −3.89380753074272646157752699044, −2.12153197851295805322541388463, −1.12365189277856981766320415323,
2.07269284386588364262983368978, 3.41501274815807021882000902292, 4.01310255652897421778086310154, 5.68901714139027001886681626589, 6.74488716783776074514539596202, 7.53116358037161721702766866574, 8.510891653852523148114162368164, 9.400558613351304971387162610754, 10.14289201944631663656647635955, 10.90331778003714722323068660065
