L(s) = 1 | + (0.320 − 0.554i)2-s + 3·3-s + (3.79 + 6.57i)4-s + 9.63·5-s + (0.960 − 1.66i)6-s + (2.30 + 3.99i)7-s + 9.98·8-s + 9·9-s + (3.08 − 5.34i)10-s + (1.77 + 3.07i)11-s + (11.3 + 19.7i)12-s + (20.9 − 36.2i)13-s + 2.95·14-s + 28.8·15-s + (−27.1 + 47.0i)16-s + (−69.1 + 119. i)17-s + ⋯ |
L(s) = 1 | + (0.113 − 0.196i)2-s + 0.577·3-s + (0.474 + 0.821i)4-s + 0.861·5-s + (0.0653 − 0.113i)6-s + (0.124 + 0.215i)7-s + 0.441·8-s + 0.333·9-s + (0.0975 − 0.168i)10-s + (0.0486 + 0.0843i)11-s + (0.273 + 0.474i)12-s + (0.446 − 0.773i)13-s + 0.0564·14-s + 0.497·15-s + (−0.424 + 0.735i)16-s + (−0.985 + 1.70i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.86500 + 0.694924i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.86500 + 0.694924i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 67 | \( 1 + (247. - 489. i)T \) |
good | 2 | \( 1 + (-0.320 + 0.554i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 9.63T + 125T^{2} \) |
| 7 | \( 1 + (-2.30 - 3.99i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-1.77 - 3.07i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-20.9 + 36.2i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (69.1 - 119. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-47.8 + 82.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-15.3 + 26.6i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (76.2 + 132. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-154. - 267. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-119. + 207. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-77.1 - 133. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 - 153.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (274. + 476. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 242.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 190.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (309. - 535. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 71 | \( 1 + (447. + 774. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-193. + 334. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (444. + 769. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (490. - 849. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 441.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-338. + 585. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.21958031261909602621514378294, −11.05066749343808258682253249344, −10.20959653660600230059096179348, −8.907860423106691469469370026112, −8.201190565422846646143618193569, −6.98472996867617103784512170810, −5.87615807726685439949320310986, −4.23769226479197869473923347192, −2.92498545430017830528182762317, −1.82806091320727085708342972962,
1.36605973313218451215086248276, 2.58638485533030423375736585175, 4.45255352373880794243827112997, 5.71851772510916363303136153273, 6.67923661340061816164306659604, 7.71579242583727706047277236122, 9.308775966924819153120475422040, 9.686471176682985915197176679702, 10.92471926019756107355092593787, 11.74234891794560858124830297648