| L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (5.49 + 3.17i)7-s − 2.82·8-s + (−8.17 − 4.71i)11-s + (17.0 − 9.84i)13-s + (7.77 − 4.48i)14-s + (−2.00 + 3.46i)16-s + 1.90·17-s − 4.69·19-s + (−11.5 + 6.67i)22-s + (−4.71 − 8.17i)23-s − 27.8i·26-s − 12.6i·28-s + (−2.84 − 1.64i)29-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.785 + 0.453i)7-s − 0.353·8-s + (−0.743 − 0.429i)11-s + (1.31 − 0.757i)13-s + (0.555 − 0.320i)14-s + (−0.125 + 0.216i)16-s + 0.112·17-s − 0.247·19-s + (−0.525 + 0.303i)22-s + (−0.205 − 0.355i)23-s − 1.07i·26-s − 0.453i·28-s + (−0.0982 − 0.0567i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.403994893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.403994893\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-5.49 - 3.17i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (8.17 + 4.71i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-17.0 + 9.84i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 1.90T + 289T^{2} \) |
| 19 | \( 1 + 4.69T + 361T^{2} \) |
| 23 | \( 1 + (4.71 + 8.17i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (2.84 + 1.64i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-20.5 - 35.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 17.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-53.5 + 30.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (0.826 + 0.477i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-7.05 + 12.2i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 9.53T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-79.2 + 45.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-37.5 + 65.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (26.8 - 15.4i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-14.8 + 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (43.9 - 76.1i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-83.0 - 47.9i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.141498976388182588512027185045, −8.408243398858170068494558454177, −7.86602800499258899982599367287, −6.50490264493176836821254379930, −5.63630921850189947563897440702, −5.02759779423375205868927886617, −3.90575141188969129158747329490, −2.98066636764265513118987847178, −1.92891697958946230784149282338, −0.68442681588998720211933911738,
1.19942917552235394371454541195, 2.54290218268725216757675500603, 3.94713187552370147013815457629, 4.48002811085262412346099885728, 5.53095051248114327543984274455, 6.29726421742338691769662584463, 7.24122417645071189841740057349, 7.953927065413700866451165995266, 8.574766146759732983832274311329, 9.544014613785412276264797187681
