| L(s) = 1 | − 2·3-s − 10·5-s − 6·9-s − 54·11-s − 26·13-s + 20·15-s + 68·17-s + 42·19-s − 294·23-s + 75·25-s − 22·27-s + 312·29-s − 170·31-s + 108·33-s − 408·37-s + 52·39-s + 140·41-s + 322·43-s + 60·45-s + 184·47-s − 366·49-s − 136·51-s − 116·53-s + 540·55-s − 84·57-s + 382·59-s + 48·61-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.894·5-s − 2/9·9-s − 1.48·11-s − 0.554·13-s + 0.344·15-s + 0.970·17-s + 0.507·19-s − 2.66·23-s + 3/5·25-s − 0.156·27-s + 1.99·29-s − 0.984·31-s + 0.569·33-s − 1.81·37-s + 0.213·39-s + 0.533·41-s + 1.14·43-s + 0.198·45-s + 0.571·47-s − 1.06·49-s − 0.373·51-s − 0.300·53-s + 1.32·55-s − 0.195·57-s + 0.842·59-s + 0.100·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 366 T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 54 T + 3266 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 p T + 9702 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 42 T - 3246 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 294 T + 45538 T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 312 T + 72134 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 170 T + 66762 T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 408 T + 137142 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 140 T + 113862 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 322 T + 183130 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 184 T + 157790 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 116 T + 22638 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 382 T + 304434 T^{2} - 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 48 T + 389558 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1620 T + 1220646 T^{2} - 1620 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 930 T + 910922 T^{2} + 930 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 609518 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 532 T + 950254 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 120 T - 163546 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 292 T + 1261974 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1092 T + 2030982 T^{2} + 1092 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−9.484037150180030795164142213445, −8.803281604424717440156401790537, −8.321451571300024782074617745286, −8.023190284934117042043632494330, −7.78583297251795207728586005888, −7.38302372188230011080586460910, −6.81666784460828010734593159054, −6.44392909475295583565965290825, −5.60679418861470381173929485949, −5.58058112300390458239118280833, −5.08829286233360168698100669777, −4.55058441939944460683303674021, −3.83455618556169685132909509682, −3.72927308451911010793484767752, −2.77728617506524408685171269366, −2.58720794803897330070629848985, −1.76441952485459531568067869515, −0.932953367968334501826826574417, 0, 0,
0.932953367968334501826826574417, 1.76441952485459531568067869515, 2.58720794803897330070629848985, 2.77728617506524408685171269366, 3.72927308451911010793484767752, 3.83455618556169685132909509682, 4.55058441939944460683303674021, 5.08829286233360168698100669777, 5.58058112300390458239118280833, 5.60679418861470381173929485949, 6.44392909475295583565965290825, 6.81666784460828010734593159054, 7.38302372188230011080586460910, 7.78583297251795207728586005888, 8.023190284934117042043632494330, 8.321451571300024782074617745286, 8.803281604424717440156401790537, 9.484037150180030795164142213445
