| L(s) = 1 | + (−3.14 − 1.14i)2-s + (−1.66 + 2.49i)3-s + (5.51 + 4.62i)4-s + (−0.611 − 4.96i)5-s + (8.09 − 5.93i)6-s + (−1.49 − 1.78i)7-s + (−5.34 − 9.25i)8-s + (−3.44 − 8.31i)9-s + (−3.75 + 16.3i)10-s + (−3.16 + 0.557i)11-s + (−20.7 + 6.04i)12-s + (6.79 + 18.6i)13-s + (2.66 + 7.32i)14-s + (13.3 + 6.74i)15-s + (1.21 + 6.88i)16-s + (−0.0949 + 0.164i)17-s + ⋯ |
| L(s) = 1 | + (−1.57 − 0.572i)2-s + (−0.555 + 0.831i)3-s + (1.37 + 1.15i)4-s + (−0.122 − 0.992i)5-s + (1.34 − 0.989i)6-s + (−0.213 − 0.254i)7-s + (−0.668 − 1.15i)8-s + (−0.383 − 0.923i)9-s + (−0.375 + 1.63i)10-s + (−0.287 + 0.0507i)11-s + (−1.72 + 0.503i)12-s + (0.522 + 1.43i)13-s + (0.190 + 0.522i)14-s + (0.893 + 0.449i)15-s + (0.0758 + 0.430i)16-s + (−0.00558 + 0.00967i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.347254 + 0.197056i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.347254 + 0.197056i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.66 - 2.49i)T \) |
| 5 | \( 1 + (0.611 + 4.96i)T \) |
| good | 2 | \( 1 + (3.14 + 1.14i)T + (3.06 + 2.57i)T^{2} \) |
| 7 | \( 1 + (1.49 + 1.78i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (3.16 - 0.557i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-6.79 - 18.6i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (0.0949 - 0.164i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-12.0 - 20.9i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-17.2 - 14.4i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (17.0 - 46.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-14.6 - 12.2i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (24.3 + 14.0i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-13.0 - 35.7i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-15.6 + 2.76i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-18.7 + 15.7i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 80.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-44.9 - 7.92i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-42.9 + 36.0i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-43.8 - 120. i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-49.0 - 28.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-99.7 + 57.6i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (13.9 + 5.07i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (66.4 + 24.1i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (108. - 62.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (157. - 27.8i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.63066808972054578406447048988, −11.69082394262823320803665468641, −10.97911027301388389568333916387, −9.861408136753269559003790878913, −9.222395867692994933309237828094, −8.387454766749951062615567387394, −6.95555405609530265432113710707, −5.25182588757928693229438730325, −3.70053758037713803959456766647, −1.31126824632366367317208111047,
0.53262629253487168061318024388, 2.62338434869930125580299988895, 5.69255345341114171500077011965, 6.62404635708735386069775378864, 7.54052785264678833123923122833, 8.283385367715122218141950108456, 9.690567475186973862607516238238, 10.76385623345335987381597009832, 11.27329729435384190363080890197, 12.74321764919115739235272398110
