| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 3·11-s − 4·13-s + 15-s + 2·21-s + 6·23-s + 25-s − 27-s + 3·29-s + 5·31-s + 3·33-s + 2·35-s + 8·37-s + 4·39-s + 6·41-s − 4·43-s − 45-s − 6·47-s − 3·49-s − 6·53-s + 3·55-s + 9·59-s + 7·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.258·15-s + 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 0.898·31-s + 0.522·33-s + 0.338·35-s + 1.31·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.824·53-s + 0.404·55-s + 1.17·59-s + 0.896·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.80112016324263, −12.43823515966316, −11.75475662349244, −11.48139989916954, −11.09755247709329, −10.41170500107477, −10.06650940527824, −9.792033864397637, −9.224009615554418, −8.639302606033023, −8.170956633115638, −7.562547095775168, −7.307072815266607, −6.768050727966050, −6.259397069949491, −5.862628005044042, −5.164249947312628, −4.705347259543715, −4.549684690672042, −3.668491096209012, −3.091280516104627, −2.709282051730666, −2.178745839681093, −1.202189726399764, −0.6134728448679586, 0,
0.6134728448679586, 1.202189726399764, 2.178745839681093, 2.709282051730666, 3.091280516104627, 3.668491096209012, 4.549684690672042, 4.705347259543715, 5.164249947312628, 5.862628005044042, 6.259397069949491, 6.768050727966050, 7.307072815266607, 7.562547095775168, 8.170956633115638, 8.639302606033023, 9.224009615554418, 9.792033864397637, 10.06650940527824, 10.41170500107477, 11.09755247709329, 11.48139989916954, 11.75475662349244, 12.43823515966316, 12.80112016324263
