| L(s) = 1 | + (0.100 + 1.41i)2-s + (−3.42 − 0.745i)3-s + (−1.97 + 0.284i)4-s + (2.66 + 4.23i)5-s + (0.706 − 4.91i)6-s + (5.71 − 10.4i)7-s + (−0.601 − 2.76i)8-s + (3.01 + 1.37i)9-s + (−5.69 + 4.18i)10-s + (1.39 + 1.61i)11-s + (7.00 + 0.500i)12-s + (9.78 + 17.9i)13-s + (15.3 + 7.01i)14-s + (−5.98 − 16.4i)15-s + (3.83 − 1.12i)16-s + (5.06 + 6.76i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 + 0.705i)2-s + (−1.14 − 0.248i)3-s + (−0.494 + 0.0711i)4-s + (0.533 + 0.846i)5-s + (0.117 − 0.818i)6-s + (0.817 − 1.49i)7-s + (−0.0751 − 0.345i)8-s + (0.334 + 0.152i)9-s + (−0.569 + 0.418i)10-s + (0.127 + 0.146i)11-s + (0.583 + 0.0417i)12-s + (0.752 + 1.37i)13-s + (1.09 + 0.500i)14-s + (−0.398 − 1.09i)15-s + (0.239 − 0.0704i)16-s + (0.297 + 0.397i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.991693 + 0.699798i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991693 + 0.699798i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.100 - 1.41i)T \) |
| 5 | \( 1 + (-2.66 - 4.23i)T \) |
| 23 | \( 1 + (-12.7 - 19.1i)T \) |
| good | 3 | \( 1 + (3.42 + 0.745i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (-5.71 + 10.4i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (-1.39 - 1.61i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-9.78 - 17.9i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (-5.06 - 6.76i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (4.21 - 0.605i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (-43.4 - 6.24i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (8.88 + 5.70i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (1.22 + 3.28i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (4.59 + 10.0i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-43.0 - 9.37i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (18.1 - 18.1i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-52.9 - 28.8i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (-13.0 + 44.5i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (74.2 + 47.7i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (2.37 + 33.1i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (-69.2 + 79.8i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-3.51 + 4.69i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (32.4 - 110. i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-84.7 + 31.6i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (-25.5 - 39.7i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-131. - 48.9i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−12.01104536161315335292338465462, −11.00938948883050569063120567434, −10.59722917859355507608289247202, −9.280584334899244057347517792036, −7.80555628304320288945582964945, −6.82090335940508761827297056422, −6.32490598042824745178818249726, −5.04593912837769922905860995118, −3.87587555854253789354525999398, −1.29757480073349263766408231854,
0.927363842003262287548826589025, 2.62858683376446502901252985210, 4.69992885609475742199850124934, 5.42641256570642073767127174395, 6.05288765516110378885142428053, 8.327560023451398547654250053197, 8.824158176929749252972434664793, 10.14694021368657057973501905438, 10.93814859051332094129284641555, 11.89968827439735714370482318586
