| L(s) = 1 | + (−1.26 − 0.642i)2-s + (5.20 + 0.823i)3-s + (1.17 + 1.61i)4-s + (−3.87 + 3.16i)5-s + (−6.02 − 4.37i)6-s + (2.45 − 2.45i)7-s + (−0.442 − 2.79i)8-s + (17.8 + 5.78i)9-s + (6.91 − 1.49i)10-s + (−4.89 − 15.0i)11-s + (4.78 + 9.38i)12-s + (−8.17 + 4.16i)13-s + (−4.66 + 1.51i)14-s + (−22.7 + 13.2i)15-s + (−1.23 + 3.80i)16-s + (−19.3 + 3.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.321i)2-s + (1.73 + 0.274i)3-s + (0.293 + 0.404i)4-s + (−0.774 + 0.632i)5-s + (−1.00 − 0.729i)6-s + (0.350 − 0.350i)7-s + (−0.0553 − 0.349i)8-s + (1.97 + 0.643i)9-s + (0.691 − 0.149i)10-s + (−0.444 − 1.36i)11-s + (0.398 + 0.782i)12-s + (−0.628 + 0.320i)13-s + (−0.332 + 0.108i)14-s + (−1.51 + 0.884i)15-s + (−0.0772 + 0.237i)16-s + (−1.13 + 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0190i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.24871 + 0.0118654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24871 + 0.0118654i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 + 0.642i)T \) |
| 5 | \( 1 + (3.87 - 3.16i)T \) |
| good | 3 | \( 1 + (-5.20 - 0.823i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-2.45 + 2.45i)T - 49iT^{2} \) |
| 11 | \( 1 + (4.89 + 15.0i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (8.17 - 4.16i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (19.3 - 3.06i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-8.31 + 11.4i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (15.4 - 30.3i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (2.27 + 3.12i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-14.0 - 10.1i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-0.500 - 0.981i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-12.1 + 37.2i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-27.1 - 27.1i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-4.98 + 31.4i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-49.7 - 7.88i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-73.2 - 23.8i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (11.5 + 35.4i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (23.4 - 3.70i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (70.8 - 51.5i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (5.28 - 10.3i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (42.3 + 58.3i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-20.8 - 131. i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-58.7 + 19.0i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (10.2 - 65.0i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−15.38308096086950931801846328527, −14.19476361503906642123519795262, −13.38825431761274572335864201778, −11.53759312674191364092146382690, −10.45392356797415264770716456196, −9.114688156007791807012565495691, −8.143501711813946881236788371721, −7.25468098758038496961344890333, −3.94738434589745245499832829118, −2.67391380409972688681203731147,
2.27643949839000462281665265954, 4.49628537075007154869959725704, 7.24963693458018032525613601211, 8.064087664320520860856464805456, 8.949252878947307449253072043044, 10.06600894581369955948986745788, 12.09124093745540412320953025403, 13.10395670320543450330226540917, 14.59628177224445780254855936724, 15.18498662436897266922061643450
