| L(s) = 1 | + (−0.590 + 2.20i)2-s + (1.05 + 1.83i)3-s + (−1.03 − 0.598i)4-s + (1.58 + 1.58i)5-s + (−4.66 + 1.24i)6-s + (−0.314 − 1.17i)7-s + (−4.51 + 4.51i)8-s + (2.26 − 3.91i)9-s + (−4.41 + 2.54i)10-s + (−13.3 − 3.57i)11-s − 2.53i·12-s + (12.5 + 3.48i)13-s + 2.76·14-s + (−1.22 + 4.57i)15-s + (−9.67 − 16.7i)16-s + (8.20 + 4.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.295 + 1.10i)2-s + (0.352 + 0.610i)3-s + (−0.259 − 0.149i)4-s + (0.316 + 0.316i)5-s + (−0.776 + 0.208i)6-s + (−0.0449 − 0.167i)7-s + (−0.564 + 0.564i)8-s + (0.251 − 0.435i)9-s + (−0.441 + 0.254i)10-s + (−1.21 − 0.324i)11-s − 0.211i·12-s + (0.963 + 0.268i)13-s + 0.197·14-s + (−0.0816 + 0.304i)15-s + (−0.604 − 1.04i)16-s + (0.482 + 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.601138 + 1.07654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.601138 + 1.07654i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.58 - 1.58i)T \) |
| 13 | \( 1 + (-12.5 - 3.48i)T \) |
| good | 2 | \( 1 + (0.590 - 2.20i)T + (-3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (-1.05 - 1.83i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (0.314 + 1.17i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (13.3 + 3.57i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (-8.20 - 4.73i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-23.4 + 6.28i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-10.0 + 5.80i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (2.30 + 3.99i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (21.8 + 21.8i)T + 961iT^{2} \) |
| 37 | \( 1 + (19.1 + 5.14i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (14.6 - 54.8i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (61.4 + 35.4i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (22.8 - 22.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 3.82T + 2.80e3T^{2} \) |
| 59 | \( 1 + (17.8 + 66.5i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-20.0 + 34.7i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (15.0 - 56.1i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (53.0 - 14.2i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-34.0 + 34.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 27.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + (20.0 + 20.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (30.1 + 8.08i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (172. - 46.1i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−15.29083692524522724239446298719, −14.29276358646704039422488825379, −13.17455628851027832578077593190, −11.48460934032780531126424912118, −10.19420882504605807471035999956, −9.044040230656589987015258224567, −7.902570485547349685137174763161, −6.65139508743130324684480115045, −5.35430575447105068213350549801, −3.25016660738582018696892862180,
1.53059359951935151756705139898, 3.06124302353320677757258034050, 5.42528728565599649862726306127, 7.25633874518101859450633578870, 8.581327824604601121937757558405, 9.919286722789345905314279772345, 10.79134351977236995206928876093, 12.11225045026067038442568511587, 13.00753449660725136167905629892, 13.76801639339210633922013325712
