| L(s) = 1 | + (−1.26 − 0.339i)2-s + (−1.96 − 1.13i)4-s + (6.74 − 1.80i)5-s + (−3.45 + 5.97i)7-s + (5.82 + 5.82i)8-s − 9.17·10-s + 1.26i·11-s + (−0.242 + 0.0649i)13-s + (6.41 − 6.41i)14-s + (−0.864 − 1.49i)16-s + (−4.52 + 16.8i)17-s + (−4.10 + 1.10i)19-s + (−15.3 − 4.11i)20-s + (0.428 − 1.60i)22-s + (19.2 + 19.2i)23-s + ⋯ |
| L(s) = 1 | + (−0.634 − 0.169i)2-s + (−0.492 − 0.284i)4-s + (1.34 − 0.361i)5-s + (−0.493 + 0.854i)7-s + (0.728 + 0.728i)8-s − 0.917·10-s + 0.114i·11-s + (−0.0186 + 0.00499i)13-s + (0.458 − 0.458i)14-s + (−0.0540 − 0.0936i)16-s + (−0.266 + 0.993i)17-s + (−0.216 + 0.0579i)19-s + (−0.767 − 0.205i)20-s + (0.0194 − 0.0727i)22-s + (0.838 + 0.838i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.13204 + 0.289233i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13204 + 0.289233i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 37 | \( 1 + (8.09 - 36.1i)T \) |
| good | 2 | \( 1 + (1.26 + 0.339i)T + (3.46 + 2i)T^{2} \) |
| 5 | \( 1 + (-6.74 + 1.80i)T + (21.6 - 12.5i)T^{2} \) |
| 7 | \( 1 + (3.45 - 5.97i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 - 1.26iT - 121T^{2} \) |
| 13 | \( 1 + (0.242 - 0.0649i)T + (146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (4.52 - 16.8i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (4.10 - 1.10i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-19.2 - 19.2i)T + 529iT^{2} \) |
| 29 | \( 1 + (-30.4 + 30.4i)T - 841iT^{2} \) |
| 31 | \( 1 + (-0.264 + 0.264i)T - 961iT^{2} \) |
| 41 | \( 1 + (-48.7 - 28.1i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-56.5 - 56.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 17.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-16.9 - 29.3i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-5.66 + 21.1i)T + (-3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (18.5 + 69.4i)T + (-3.22e3 + 1.86e3i)T^{2} \) |
| 67 | \( 1 + (-30.0 - 17.3i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (18.1 - 31.3i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + 108. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (98.0 - 26.2i)T + (5.40e3 - 3.12e3i)T^{2} \) |
| 83 | \( 1 + (9.22 + 15.9i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-123. - 33.0i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (129. + 129. i)T + 9.40e3iT^{2} \) |
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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
−11.15871689475177242311978691302, −10.17554378278779667921017096275, −9.489515822603453054629529126929, −8.970876050907774874120055548288, −7.986895208601570872198096776528, −6.28790614833207065654008056142, −5.66141781410343181641819882307, −4.54086927146609320552914828366, −2.55616157795442937301133773526, −1.33279718976974992281944542093,
0.78209679800974397041604827297, 2.65236183490305825774568085723, 4.12500839459058640812874247975, 5.40586249916520052249314410002, 6.74686349722374261763301610632, 7.29247770694298382076725344633, 8.778647579314694430986128117822, 9.331925687013736030104125914728, 10.30943963399222706064997607211, 10.72190960589341425964048504510
