L(s) = 1 | − 2.68·2-s + (−3.19 − 4.09i)3-s − 0.797·4-s − 15.8i·5-s + (8.57 + 10.9i)6-s + 25.4·7-s + 23.6·8-s + (−6.56 + 26.1i)9-s + 42.4i·10-s + 64.0·11-s + (2.54 + 3.26i)12-s − 26.7i·13-s − 68.2·14-s + (−64.7 + 50.5i)15-s − 56.9·16-s − 54.3i·17-s + ⋯ |
L(s) = 1 | − 0.948·2-s + (−0.615 − 0.788i)3-s − 0.0997·4-s − 1.41i·5-s + (0.583 + 0.748i)6-s + 1.37·7-s + 1.04·8-s + (−0.243 + 0.969i)9-s + 1.34i·10-s + 1.75·11-s + (0.0613 + 0.0786i)12-s − 0.570i·13-s − 1.30·14-s + (−1.11 + 0.870i)15-s − 0.890·16-s − 0.775i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.553007 - 0.824152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.553007 - 0.824152i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.19 + 4.09i)T \) |
| 59 | \( 1 + (224. + 393. i)T \) |
good | 2 | \( 1 + 2.68T + 8T^{2} \) |
| 5 | \( 1 + 15.8iT - 125T^{2} \) |
| 7 | \( 1 - 25.4T + 343T^{2} \) |
| 11 | \( 1 - 64.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 54.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 47.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 169. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 217. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 122. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 368. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 3.01iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 468. iT - 1.48e5T^{2} \) |
| 61 | \( 1 - 558. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 284. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 279. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 877. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 63.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 941.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 199.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 432. iT - 9.12e5T^{2} \) |
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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
−11.73015360720744894087590131911, −11.16946808069488580343696995255, −9.628994796468105626912777041807, −8.712380803160626519035091123653, −8.076020738579953270896537087284, −6.98899084924120852012300792816, −5.25052671533165553212238628035, −4.59770877983175183703174968110, −1.40962850918489519539349617188, −0.914060490225708671213291151452,
1.40190158591476565488047956375, 3.71629282806275236748595991793, 4.83160369748694343393772073424, 6.43103679271381678239845710426, 7.39461246665669806985459732254, 8.801365763035485472525553185754, 9.504568354903394895001251819936, 10.74787340756897929129883020109, 11.06481168241411059381957822857, 11.95468441273959188314870702056