| L(s) = 1 | − 8·2-s + 18·3-s + 48·4-s − 49·5-s − 144·6-s − 105·7-s − 256·8-s + 243·9-s + 392·10-s + 725·11-s + 864·12-s − 56·13-s + 840·14-s − 882·15-s + 1.28e3·16-s + 1.52e3·17-s − 1.94e3·18-s − 722·19-s − 2.35e3·20-s − 1.89e3·21-s − 5.80e3·22-s + 1.70e3·23-s − 4.60e3·24-s − 3.42e3·25-s + 448·26-s + 2.91e3·27-s − 5.04e3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.876·5-s − 1.63·6-s − 0.809·7-s − 1.41·8-s + 9-s + 1.23·10-s + 1.80·11-s + 1.73·12-s − 0.0919·13-s + 1.14·14-s − 1.01·15-s + 5/4·16-s + 1.27·17-s − 1.41·18-s − 0.458·19-s − 1.31·20-s − 0.935·21-s − 2.55·22-s + 0.670·23-s − 1.63·24-s − 1.09·25-s + 0.129·26-s + 0.769·27-s − 1.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.213547065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213547065\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 49 T + 5828 T^{2} + 49 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 15 p T + 10814 T^{2} + 15 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 725 T + 452486 T^{2} - 725 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 56 T + 334470 T^{2} + 56 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1521 T + 1863232 T^{2} - 1521 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1700 T - 160210 T^{2} - 1700 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12724 T + 81235646 T^{2} - 12724 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8558 T + 58292118 T^{2} - 8558 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12434 T + 135627114 T^{2} - 12434 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 20230 T + 259503602 T^{2} - 20230 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4895 T + 299556330 T^{2} - 4895 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21059 T + 555084302 T^{2} - 21059 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 25196 T + 966248606 T^{2} - 25196 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1560 T + 1429410214 T^{2} - 1560 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2123 T - 588325956 T^{2} + 2123 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 46968 T + 3250701686 T^{2} + 46968 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 60056 T + 2271003086 T^{2} - 60056 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 57177 T + 4702746212 T^{2} + 57177 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 99368 T + 8517934254 T^{2} + 99368 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 21780 T - 3626701658 T^{2} + 21780 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11856 T + 2415361198 T^{2} + 11856 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 77068 T - 559666266 T^{2} + 77068 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.59077662442301502535017374809, −12.34699905643317480108195750971, −11.66689302118222171502422080708, −11.53877921073748755680088974846, −10.38404933270753762296681404857, −10.18348210530864781605951302574, −9.532086136105657560223432370297, −9.191725621198016247111052480499, −8.583966024772736185898767506845, −8.196511280388750974797120557625, −7.56581412496507055927749357261, −7.15622885548722860066789836077, −6.37570646886252990644723498936, −6.03295358005877565783144992649, −4.34245187045827182781694863796, −3.97500465481077572317087483570, −2.99102723988132078536658953077, −2.57445413174859067452073381954, −1.15250050606863482145280041400, −0.812121560386603378916570811270,
0.812121560386603378916570811270, 1.15250050606863482145280041400, 2.57445413174859067452073381954, 2.99102723988132078536658953077, 3.97500465481077572317087483570, 4.34245187045827182781694863796, 6.03295358005877565783144992649, 6.37570646886252990644723498936, 7.15622885548722860066789836077, 7.56581412496507055927749357261, 8.196511280388750974797120557625, 8.583966024772736185898767506845, 9.191725621198016247111052480499, 9.532086136105657560223432370297, 10.18348210530864781605951302574, 10.38404933270753762296681404857, 11.53877921073748755680088974846, 11.66689302118222171502422080708, 12.34699905643317480108195750971, 12.59077662442301502535017374809
