| L(s) = 1 | − 2·3-s − 6·5-s − 56·7-s − 31·9-s + 22·11-s − 44·13-s + 12·15-s − 8·17-s − 188·19-s + 112·21-s − 66·23-s + 97·25-s + 78·27-s − 56·29-s + 206·31-s − 44·33-s + 336·35-s + 142·37-s + 88·39-s + 252·41-s − 372·43-s + 186·45-s − 200·47-s + 1.68e3·49-s + 16·51-s − 1.06e3·53-s − 132·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.536·5-s − 3.02·7-s − 1.14·9-s + 0.603·11-s − 0.938·13-s + 0.206·15-s − 0.114·17-s − 2.27·19-s + 1.16·21-s − 0.598·23-s + 0.775·25-s + 0.555·27-s − 0.358·29-s + 1.19·31-s − 0.232·33-s + 1.62·35-s + 0.630·37-s + 0.361·39-s + 0.959·41-s − 1.31·43-s + 0.616·45-s − 0.620·47-s + 4.91·49-s + 0.0439·51-s − 2.74·53-s − 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 35 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 6 T - 61 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8 p T + 1450 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 44 T + 2458 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 5342 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 188 T + 20934 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 66 T + 8603 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 56 T + 7242 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 206 T + 69211 T^{2} - 206 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 142 T + 52267 T^{2} - 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 252 T + 150338 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 372 T + 177930 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 200 T + 211166 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 20 p T + 574734 T^{2} + 20 p^{4} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 782 T + 519459 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 240 T + 370362 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 282 T + 536907 T^{2} - 282 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 754 T + 853451 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 68 T + 260770 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 52 T + 831874 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 592458 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 278 T + 737339 T^{2} - 278 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 306 T + 1632435 T^{2} - 306 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.04913727363512579253475579392, −12.90795491790289129567580082843, −12.32649442468505467200883865058, −12.07098254741489874733056527222, −11.16567476976165648613187736505, −10.77639666704985673478170697072, −9.924079464444527899667728647718, −9.659122053155969558081958988247, −9.057593485842053272379405956051, −8.436743048547681304540774827586, −7.67024011723645141427008216088, −6.62095582438664376361297220394, −6.30704954846364569046733495020, −6.21876328928315258291377157276, −4.88809635308001568501547299964, −3.99096818031750137711652044456, −3.19094457076919009030847200644, −2.59304597539575936745848933381, 0, 0,
2.59304597539575936745848933381, 3.19094457076919009030847200644, 3.99096818031750137711652044456, 4.88809635308001568501547299964, 6.21876328928315258291377157276, 6.30704954846364569046733495020, 6.62095582438664376361297220394, 7.67024011723645141427008216088, 8.436743048547681304540774827586, 9.057593485842053272379405956051, 9.659122053155969558081958988247, 9.924079464444527899667728647718, 10.77639666704985673478170697072, 11.16567476976165648613187736505, 12.07098254741489874733056527222, 12.32649442468505467200883865058, 12.90795491790289129567580082843, 13.04913727363512579253475579392
