| L(s) = 1 | − i·2-s − 2.29i·3-s − 4-s + (1.14 + 1.91i)5-s − 2.29·6-s + 1.25i·7-s + i·8-s − 2.25·9-s + (1.91 − 1.14i)10-s − 2·11-s + 2.29i·12-s + 1.25·14-s + (4.40 − 2.62i)15-s + 16-s + 4.80i·17-s + 2.25i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.32i·3-s − 0.5·4-s + (0.512 + 0.858i)5-s − 0.935·6-s + 0.474i·7-s + 0.353i·8-s − 0.751·9-s + (0.607 − 0.362i)10-s − 0.603·11-s + 0.661i·12-s + 0.335·14-s + (1.13 − 0.678i)15-s + 0.250·16-s + 1.16i·17-s + 0.531i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9616835624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9616835624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (-1.14 - 1.91i)T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.29iT - 3T^{2} \) |
| 7 | \( 1 - 1.25iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 4.80iT - 17T^{2} \) |
| 19 | \( 1 + 5.09T + 19T^{2} \) |
| 23 | \( 1 - 2.58iT - 23T^{2} \) |
| 29 | \( 1 + 5.09T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 7.83iT - 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 - 2.74iT - 47T^{2} \) |
| 53 | \( 1 - 2.58iT - 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 - 8.58iT - 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 11.0iT - 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 + 6.26iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.480038383759947459258206338703, −8.571608372290252940676374279754, −7.78198352266601102032665914827, −7.09519561755741142131961778720, −6.08467748035738281168799345295, −5.70988132982080255317919501580, −4.23211410913785260343198781683, −3.04549089364469482585501851863, −2.19665843628802814314671928669, −1.53881020815766279284031490392,
0.34914101344907292780115050843, 2.26563460952745448349214025586, 3.82427051029263443529688038497, 4.34182164446629490977940824207, 5.29160798686814381223344505510, 5.62826288453450062058658546885, 6.93362293875076357150746426747, 7.70467867206141201307851761679, 8.838918564077556610265515273473, 9.107036721628647692425105054097
