| L(s) = 1 | + 12·2-s − 21·3-s + 64·4-s + 315·5-s − 252·6-s − 686·7-s + 576·8-s − 435·9-s + 3.78e3·10-s − 1.47e3·11-s − 1.34e3·12-s − 8.23e3·14-s − 6.61e3·15-s + 6.91e3·16-s − 5.22e3·17-s − 5.22e3·18-s + 1.19e4·19-s + 2.01e4·20-s + 1.44e4·21-s − 1.77e4·22-s + 5.91e3·23-s − 1.20e4·24-s + 5.05e4·25-s + 1.22e4·27-s − 4.39e4·28-s + 7.95e3·29-s − 7.93e4·30-s + ⋯ |
| L(s) = 1 | + 3/2·2-s − 7/9·3-s + 4-s + 2.51·5-s − 7/6·6-s − 2·7-s + 9/8·8-s − 0.596·9-s + 3.77·10-s − 1.11·11-s − 7/9·12-s − 3·14-s − 1.95·15-s + 1.68·16-s − 1.06·17-s − 0.895·18-s + 1.73·19-s + 2.51·20-s + 14/9·21-s − 1.66·22-s + 0.485·23-s − 7/8·24-s + 3.23·25-s + 0.620·27-s − 2·28-s + 0.326·29-s − 2.93·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.379731248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.379731248\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p^{2} T + 5 p^{4} T^{2} - 3 p^{8} T^{3} + p^{12} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 7 p T + 292 p T^{2} + 7 p^{7} T^{3} + p^{12} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 63 p T + 1948 p^{2} T^{2} - 63 p^{7} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 1479 T + 415880 T^{2} + 1479 p^{6} T^{3} + p^{12} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 9418418 T^{2} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5229 T + 33251716 T^{2} + 5229 p^{6} T^{3} + p^{12} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 11907 T + 94304764 T^{2} - 11907 p^{6} T^{3} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 5913 T - 113072320 T^{2} - 5913 p^{6} T^{3} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3978 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 22197 T + 1051739284 T^{2} + 22197 p^{6} T^{3} + p^{12} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 61577 T + 1226000520 T^{2} - 61577 p^{6} T^{3} + p^{12} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2726428318 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 17414 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 53109 T + 11719403956 T^{2} + 53109 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 60513 T - 18502537960 T^{2} - 60513 p^{6} T^{3} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 373653 T + 88719388444 T^{2} + 373653 p^{6} T^{3} + p^{12} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 281883 T + 78006382924 T^{2} - 281883 p^{6} T^{3} + p^{12} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 268777 T - 18217306440 T^{2} - 268777 p^{6} T^{3} + p^{12} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 101922 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 550179 T + 252233203636 T^{2} - 550179 p^{6} T^{3} + p^{12} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 362231 T - 111876158160 T^{2} + 362231 p^{6} T^{3} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 606885669938 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2311533 T + 2278042894324 T^{2} + 2311533 p^{6} T^{3} + p^{12} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 626523106942 T^{2} + p^{12} T^{4} \) |
| 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
−22.05933765386141182914082004724, −21.42200153598462369562707372982, −20.38002111999181750100963248619, −19.76607650522534247978860423039, −18.41341380025591701176628862167, −17.84228833585390779634555733243, −16.86224901203694508411138291154, −16.50111391846880863031818652137, −15.49348055359627014559570455228, −14.02163605348064336224635231819, −13.70429933300513078472115009033, −13.06725590923923553274413397812, −12.68512126154162032588112741914, −11.03921504661519098143836210849, −10.02749283441153501165187102793, −9.457427395277246449607015568183, −6.73625150965515819440507321704, −5.74983451610027886989667333927, −5.35236797248870929593580240766, −2.79702464582799503960219017120,
2.79702464582799503960219017120, 5.35236797248870929593580240766, 5.74983451610027886989667333927, 6.73625150965515819440507321704, 9.457427395277246449607015568183, 10.02749283441153501165187102793, 11.03921504661519098143836210849, 12.68512126154162032588112741914, 13.06725590923923553274413397812, 13.70429933300513078472115009033, 14.02163605348064336224635231819, 15.49348055359627014559570455228, 16.50111391846880863031818652137, 16.86224901203694508411138291154, 17.84228833585390779634555733243, 18.41341380025591701176628862167, 19.76607650522534247978860423039, 20.38002111999181750100963248619, 21.42200153598462369562707372982, 22.05933765386141182914082004724
