L(s) = 1 | + (−0.755 + 0.436i)2-s + (−2.59 − 1.5i)3-s + (−3.61 + 6.26i)4-s + (−7.84 + 7.96i)5-s + 2.61·6-s + (10.7 − 15.0i)7-s − 13.2i·8-s + (4.5 + 7.79i)9-s + (2.45 − 9.43i)10-s + (10.5 − 18.2i)11-s + (18.8 − 10.8i)12-s − 62.2i·13-s + (−1.56 + 16.0i)14-s + (32.3 − 8.92i)15-s + (−23.1 − 40.1i)16-s + (−35.9 − 20.7i)17-s + ⋯ |
L(s) = 1 | + (−0.267 + 0.154i)2-s + (−0.499 − 0.288i)3-s + (−0.452 + 0.783i)4-s + (−0.701 + 0.712i)5-s + 0.178·6-s + (0.581 − 0.813i)7-s − 0.587i·8-s + (0.166 + 0.288i)9-s + (0.0775 − 0.298i)10-s + (0.288 − 0.500i)11-s + (0.452 − 0.261i)12-s − 1.32i·13-s + (−0.0298 + 0.306i)14-s + (0.556 − 0.153i)15-s + (−0.361 − 0.626i)16-s + (−0.512 − 0.295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.439176 - 0.393778i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.439176 - 0.393778i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.59 + 1.5i)T \) |
| 5 | \( 1 + (7.84 - 7.96i)T \) |
| 7 | \( 1 + (-10.7 + 15.0i)T \) |
good | 2 | \( 1 + (0.755 - 0.436i)T + (4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-10.5 + 18.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 62.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (35.9 + 20.7i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-11.2 - 19.4i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (33.8 - 19.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 59.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-152. + 264. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-138. + 79.8i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 497.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 257. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (446. - 257. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-36.1 - 20.8i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-136. + 235. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (371. + 642. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (661. + 381. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 42.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-104. - 60.3i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-345. - 598. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.23e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-412. - 715. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 259. iT - 9.12e5T^{2} \) |
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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
−13.05844541482908455385050523993, −11.84636665462515337052520784149, −11.07624572348093926667295219799, −9.945645958428919157294165631471, −8.156972433927172014153780997891, −7.70015273437671719507148065653, −6.46296953818860698764837100949, −4.59238480709626489708888667364, −3.29461591958714745265230932526, −0.40946552563234894495004703263,
1.57109661894033448810632447491, 4.40051748918715451597107232900, 5.12412545445482099214066531617, 6.61971487970322619622458071386, 8.481796617830651153151912248014, 9.111380791786340642239911570114, 10.31671899886057643237148961259, 11.60504796028288299813335257833, 12.03514971922719618627256110263, 13.54577929657338290476275921053