| L(s) = 1 | + 2.23i·2-s + i·3-s − 3.00·4-s − 2.23·6-s − 1.23i·7-s − 2.23i·8-s − 9-s + 4·11-s − 3.00i·12-s − 4.47i·13-s + 2.76·14-s − 0.999·16-s − 7.23i·17-s − 2.23i·18-s − 2.76·19-s + ⋯ |
| L(s) = 1 | + 1.58i·2-s + 0.577i·3-s − 1.50·4-s − 0.912·6-s − 0.467i·7-s − 0.790i·8-s − 0.333·9-s + 1.20·11-s − 0.866i·12-s − 1.24i·13-s + 0.738·14-s − 0.249·16-s − 1.75i·17-s − 0.527i·18-s − 0.634·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.443837975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.443837975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 7 | \( 1 + 1.23iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 7.23iT - 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 + 7.70iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 0.763iT - 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 - 5.23iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10.9iT - 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 + 0.472iT - 97T^{2} \) |
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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
−9.133185732387145543653524606614, −8.660156673645313011450632540593, −7.68238145308441289298750160729, −7.10296507714916500455330856764, −6.29238048579079674750426211460, −5.52588483379597859265363300618, −4.70531552802819197822012897379, −3.98878072031485240616312845926, −2.74496564587504473954070585975, −0.60150836753872770240475311050,
1.30315344979837630629439168600, 1.89356890242984485891384577630, 2.95981736922249806618260957663, 4.01554691251204851352552213491, 4.57326434877802294979288167487, 6.22986674859221491759671636237, 6.46857622256113892840703183841, 7.83227834388023371009648124382, 8.837648803367975816980925667345, 9.147198588994774952576070701796
