L(s) = 1 | − 2·2-s − 4-s + 10·5-s + 8·7-s − 4·8-s − 20·10-s − 130·13-s − 16·14-s − 19·16-s − 14·17-s + 48·19-s − 10·20-s + 128·23-s − 163·25-s + 260·26-s − 8·28-s − 30·29-s − 184·31-s + 202·32-s + 28·34-s + 80·35-s + 126·37-s − 96·38-s − 40·40-s + 370·41-s + 264·43-s − 256·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/8·4-s + 0.894·5-s + 0.431·7-s − 0.176·8-s − 0.632·10-s − 2.77·13-s − 0.305·14-s − 0.296·16-s − 0.199·17-s + 0.579·19-s − 0.111·20-s + 1.16·23-s − 1.30·25-s + 1.96·26-s − 0.0539·28-s − 0.192·29-s − 1.06·31-s + 1.11·32-s + 0.141·34-s + 0.386·35-s + 0.559·37-s − 0.409·38-s − 0.158·40-s + 1.40·41-s + 0.936·43-s − 0.820·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.393712708\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.393712708\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
good | 2 | $D_{4}$ | \( 1 + p T + 5 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 p T + 263 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 8 T + 114 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 p T + 8607 T^{2} + 10 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 14 T + 5987 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 48 T + 13322 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 128 T + 15362 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 30 T + 32575 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 184 T + 41538 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 126 T + 30383 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 370 T + 172019 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 264 T + 170630 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 256 T + 76178 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 162 T + 217615 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1304 T + 814694 T^{2} - 1304 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 300 T + 374894 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 656 T + 707658 T^{2} + 656 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1176 T + 1023934 T^{2} - 1176 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 668 T + 500790 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 416 T + 886770 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 960 T + 1342762 T^{2} + 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1074 T + 1326595 T^{2} - 1074 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 338 T + 1196835 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.537691773393255577006059494258, −9.517046589720918422743302160166, −9.022530588374038581423778626165, −8.613988810527275089177050912487, −7.88723344479294634915884856093, −7.79258622352720849562537900159, −7.16579890593403550827398929611, −6.99866176008144604398226026610, −6.43210255016540815738082777003, −5.65519685650764544417367188193, −5.52231453465964332953839396420, −5.04985599893705187566762006323, −4.53998703557849889106836395315, −4.12397690063656553624817031925, −3.32561571731497337920574766434, −2.60142248845277907894792962417, −2.25781248779417798522424691039, −1.93623153677733689207418339686, −0.864761260769871819672149775817, −0.40598454879212157604919643259,
0.40598454879212157604919643259, 0.864761260769871819672149775817, 1.93623153677733689207418339686, 2.25781248779417798522424691039, 2.60142248845277907894792962417, 3.32561571731497337920574766434, 4.12397690063656553624817031925, 4.53998703557849889106836395315, 5.04985599893705187566762006323, 5.52231453465964332953839396420, 5.65519685650764544417367188193, 6.43210255016540815738082777003, 6.99866176008144604398226026610, 7.16579890593403550827398929611, 7.79258622352720849562537900159, 7.88723344479294634915884856093, 8.613988810527275089177050912487, 9.022530588374038581423778626165, 9.517046589720918422743302160166, 9.537691773393255577006059494258