| L(s) = 1 | + (0.683 + 0.472i)2-s + (−0.885 + 0.464i)3-s + (−0.464 − 1.22i)4-s + (−0.740 + 0.836i)5-s + (−0.824 − 0.100i)6-s + (−0.471 + 1.91i)7-s + (0.658 − 2.67i)8-s + (0.568 − 0.822i)9-s + (−0.901 + 0.222i)10-s + (−3.09 + 2.13i)11-s + (0.980 + 0.868i)12-s + (−3.00 + 1.99i)13-s + (−1.22 + 1.08i)14-s + (0.267 − 1.08i)15-s + (−0.249 + 0.221i)16-s + (−2.25 − 0.556i)17-s + ⋯ |
| L(s) = 1 | + (0.483 + 0.333i)2-s + (−0.511 + 0.268i)3-s + (−0.232 − 0.612i)4-s + (−0.331 + 0.373i)5-s + (−0.336 − 0.0408i)6-s + (−0.178 + 0.723i)7-s + (0.232 − 0.944i)8-s + (0.189 − 0.274i)9-s + (−0.285 + 0.0702i)10-s + (−0.932 + 0.643i)11-s + (0.282 + 0.250i)12-s + (−0.832 + 0.554i)13-s + (−0.327 + 0.290i)14-s + (0.0690 − 0.280i)15-s + (−0.0624 + 0.0553i)16-s + (−0.547 − 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0343310 + 0.418486i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0343310 + 0.418486i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.00 - 1.99i)T \) |
| good | 2 | \( 1 + (-0.683 - 0.472i)T + (0.709 + 1.87i)T^{2} \) |
| 5 | \( 1 + (0.740 - 0.836i)T + (-0.602 - 4.96i)T^{2} \) |
| 7 | \( 1 + (0.471 - 1.91i)T + (-6.19 - 3.25i)T^{2} \) |
| 11 | \( 1 + (3.09 - 2.13i)T + (3.90 - 10.2i)T^{2} \) |
| 17 | \( 1 + (2.25 + 0.556i)T + (15.0 + 7.90i)T^{2} \) |
| 19 | \( 1 + 0.681iT - 19T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 + (2.17 - 3.15i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (-0.413 - 0.0502i)T + (30.0 + 7.41i)T^{2} \) |
| 37 | \( 1 + (2.75 + 0.335i)T + (35.9 + 8.85i)T^{2} \) |
| 41 | \( 1 + (3.87 + 7.38i)T + (-23.2 + 33.7i)T^{2} \) |
| 43 | \( 1 + (-0.129 - 1.06i)T + (-41.7 + 10.2i)T^{2} \) |
| 47 | \( 1 + (7.89 + 2.99i)T + (35.1 + 31.1i)T^{2} \) |
| 53 | \( 1 + (-11.6 - 2.87i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (3.50 - 3.95i)T + (-7.11 - 58.5i)T^{2} \) |
| 61 | \( 1 + (-13.9 + 3.43i)T + (54.0 - 28.3i)T^{2} \) |
| 67 | \( 1 + (-3.93 - 1.49i)T + (50.1 + 44.4i)T^{2} \) |
| 71 | \( 1 + (-2.85 - 5.44i)T + (-40.3 + 58.4i)T^{2} \) |
| 73 | \( 1 + (-5.72 + 3.95i)T + (25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-0.277 + 0.731i)T + (-59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (1.19 - 2.28i)T + (-47.1 - 68.3i)T^{2} \) |
| 89 | \( 1 - 0.226iT - 89T^{2} \) |
| 97 | \( 1 + (10.8 + 12.1i)T + (-11.6 + 96.2i)T^{2} \) |
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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
−11.32735389799651275320353301022, −10.36843459679749803010077065984, −9.740875127237530462122730802061, −8.813822450456208818851399836849, −7.34681547652734983506545323231, −6.69377528388962479705403342747, −5.51769523928778776316241922756, −4.97581099573878374614276772079, −3.85973330059001961755923762735, −2.21766533457630723909575082921,
0.21372161197728566324815999926, 2.43608729899519046869612057985, 3.71458930507660943628756473006, 4.67500348254802929045895559376, 5.54616785880936528725988134944, 6.86836484042543226375722153617, 7.944628969067545256851634925850, 8.322368484335562807395251872179, 9.854525328787747400794525090243, 10.67106896760536185577992626241
