L(s) = 1 | − i·2-s − i·3-s − 4-s + 5-s − 6-s − 2·7-s + i·8-s + 2·9-s − i·10-s + 5i·11-s + i·12-s − 13-s + 2i·14-s − i·15-s + 16-s + 2i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353i·8-s + 0.666·9-s − 0.316i·10-s + 1.50i·11-s + 0.288i·12-s − 0.277·13-s + 0.534i·14-s − 0.258i·15-s + 0.250·16-s + 0.485i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.705123 - 0.477391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.705123 - 0.477391i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 29 | \( 1 + (-5 + 2i)T \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 5iT - 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 31 | \( 1 + 5iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 - iT - 79T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + 14iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−14.83758216261447345291267044001, −13.41184115269456148166006138874, −12.79223661837676310690463501944, −11.84645776206177216181121347730, −10.11355450344949753919883085106, −9.574900076543553179030084851064, −7.69276887016559049573974177703, −6.35874181585401900380951870048, −4.37322746476470901250864601121, −2.15684754175300937699526679967,
3.67787809784757201819998183332, 5.46741702281727274032318267978, 6.66752135726560757641321157411, 8.304437480803866732936226297063, 9.606145269789884566896974019170, 10.42310480291414364991838362568, 12.19159594137950504415190357871, 13.52602251868703231743237573621, 14.26119778816102661935931644068, 15.87128460803352451050194612739