L(s) = 1 | + 9.26i·3-s − 16.4i·5-s − 34.5i·7-s − 58.7·9-s − 7.42i·11-s − 42.0·13-s + 151.·15-s + (−26.0 − 65.0i)17-s + 59.9·19-s + 319.·21-s − 49.4i·23-s − 144.·25-s − 294. i·27-s + 259. i·29-s + 92.2i·31-s + ⋯ |
L(s) = 1 | + 1.78i·3-s − 1.46i·5-s − 1.86i·7-s − 2.17·9-s − 0.203i·11-s − 0.896·13-s + 2.61·15-s + (−0.372 − 0.928i)17-s + 0.723·19-s + 3.32·21-s − 0.448i·23-s − 1.15·25-s − 2.09i·27-s + 1.66i·29-s + 0.534i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.915480 - 0.619244i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.915480 - 0.619244i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (26.0 + 65.0i)T \) |
good | 3 | \( 1 - 9.26iT - 27T^{2} \) |
| 5 | \( 1 + 16.4iT - 125T^{2} \) |
| 7 | \( 1 + 34.5iT - 343T^{2} \) |
| 11 | \( 1 + 7.42iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 42.0T + 2.19e3T^{2} \) |
| 19 | \( 1 - 59.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 259. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 92.2iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 207. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 176. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 19.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 80.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 319.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 11.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 712. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 484.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 443. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 337. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 840. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 456.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 638. iT - 9.12e5T^{2} \) |
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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
−12.53315395242571226980320122948, −11.22362191270355462942603810655, −10.34339968845934339060030177574, −9.530148666255956401861039080227, −8.705614401392361533952211589774, −7.29036317821476901918417651024, −5.15093853223361305431554412119, −4.63310322515273990066849221236, −3.56600680122728331184887710562, −0.53926807483611096241002500927,
2.11174470696619767331386013847, 2.80061386230783044682825253551, 5.71400502052104329540434254131, 6.44669743628292844935642238731, 7.45461853698736173845145159682, 8.383990901346061204934039108047, 9.780091063381266607234291876389, 11.43720668445035139650945934400, 11.89561172547254941015719182888, 12.82938778358412499500677169745