| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s + 5-s + 6·6-s − 2·7-s − 3·8-s + 3·9-s − 3·10-s − 8·12-s − 4·13-s + 6·14-s − 2·15-s + 3·16-s − 9·17-s − 9·18-s + 5·19-s + 4·20-s + 4·21-s − 4·23-s + 6·24-s − 8·25-s + 12·26-s − 4·27-s − 8·28-s − 12·29-s + 6·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s + 0.447·5-s + 2.44·6-s − 0.755·7-s − 1.06·8-s + 9-s − 0.948·10-s − 2.30·12-s − 1.10·13-s + 1.60·14-s − 0.516·15-s + 3/4·16-s − 2.18·17-s − 2.12·18-s + 1.14·19-s + 0.894·20-s + 0.872·21-s − 0.834·23-s + 1.22·24-s − 8/5·25-s + 2.35·26-s − 0.769·27-s − 1.51·28-s − 2.22·29-s + 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 85 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 81 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 83 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 107 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17 T + 189 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 125 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 87 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 209 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 203 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.87953420496836069222112139210, −10.78289067142785881077653490406, −9.863709486678904235572054203793, −9.827318863042998182734568494876, −9.396881642356561666798327339601, −9.347426769669901067521771697937, −8.313693641954583186886394218883, −8.192667823969700711798817326454, −7.36964410241537205204447272507, −7.08970735101694332655335795573, −6.57840427014371539352422085539, −6.05540157654284710868464828254, −5.42086985009428714237368537543, −5.03013832220991572063030127937, −4.08942123162730063370882791638, −3.41652021169201269082197428946, −2.14593612553279308138501529502, −1.69987514028985068024643792656, 0, 0,
1.69987514028985068024643792656, 2.14593612553279308138501529502, 3.41652021169201269082197428946, 4.08942123162730063370882791638, 5.03013832220991572063030127937, 5.42086985009428714237368537543, 6.05540157654284710868464828254, 6.57840427014371539352422085539, 7.08970735101694332655335795573, 7.36964410241537205204447272507, 8.192667823969700711798817326454, 8.313693641954583186886394218883, 9.347426769669901067521771697937, 9.396881642356561666798327339601, 9.827318863042998182734568494876, 9.863709486678904235572054203793, 10.78289067142785881077653490406, 10.87953420496836069222112139210
