| L(s) = 1 | + 4·2-s − 84·5-s + 193·7-s − 64·8-s − 336·10-s − 348·11-s − 845·13-s + 772·14-s − 256·16-s + 3.38e3·17-s − 158·19-s − 1.39e3·22-s + 564·23-s + 3.12e3·25-s − 3.38e3·26-s − 6.43e3·29-s − 4.94e3·31-s + 1.35e4·34-s − 1.62e4·35-s − 7.61e3·37-s − 632·38-s + 5.37e3·40-s + 1.24e4·41-s + 4.93e3·43-s + 2.25e3·46-s − 8.12e3·47-s + 1.68e4·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.50·5-s + 1.48·7-s − 0.353·8-s − 1.06·10-s − 0.867·11-s − 1.38·13-s + 1.05·14-s − 1/4·16-s + 2.83·17-s − 0.100·19-s − 0.613·22-s + 0.222·23-s + 25-s − 0.980·26-s − 1.42·29-s − 0.923·31-s + 2.00·34-s − 2.23·35-s − 0.913·37-s − 0.0709·38-s + 0.531·40-s + 1.15·41-s + 0.407·43-s + 0.157·46-s − 0.536·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.665610146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665610146\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 84 T + 3931 T^{2} + 84 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 193 T + 20442 T^{2} - 193 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 348 T - 39947 T^{2} + 348 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 p^{2} T + p^{5} T^{2} )( 1 + 7 p^{2} T + p^{5} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 1692 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 79 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 564 T - 6118247 T^{2} - 564 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6432 T + 20859475 T^{2} + 6432 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 4940 T - 4225551 T^{2} + 4940 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 3805 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12480 T + 39894199 T^{2} - 12480 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4936 T - 122644347 T^{2} - 4936 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8124 T - 163345631 T^{2} + 8124 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 33192 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 42492 T + 1090645765 T^{2} + 42492 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 17833 T - 526580412 T^{2} - 17833 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 67699 T + 3233029494 T^{2} - 67699 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 28152 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 13975 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 83983 T + 3976087890 T^{2} - 83983 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 33384 T - 2824549187 T^{2} - 33384 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 77868 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2083 T - 8583001368 T^{2} - 2083 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.33205745521536393189058579849, −11.70674144548439774937893352603, −11.43032821121379992481253538533, −10.82972934044981162317480867543, −10.45799274715920394230507479839, −9.524658547029998057303092263148, −9.343037526212950437127966657194, −8.144580825525410916708064312906, −7.991002524072681410804448304411, −7.54231411997539342537513866192, −7.38157176597654075804422861509, −6.18465977482245231741357449418, −5.28276279248139310880371516968, −5.11367343999587375176832523962, −4.65062398939425706287788215024, −3.59174033401251007677310682243, −3.47947537461102542529249558553, −2.39325242324356636907519418648, −1.43104433408985254881241000512, −0.39073262820449087411854074724,
0.39073262820449087411854074724, 1.43104433408985254881241000512, 2.39325242324356636907519418648, 3.47947537461102542529249558553, 3.59174033401251007677310682243, 4.65062398939425706287788215024, 5.11367343999587375176832523962, 5.28276279248139310880371516968, 6.18465977482245231741357449418, 7.38157176597654075804422861509, 7.54231411997539342537513866192, 7.991002524072681410804448304411, 8.144580825525410916708064312906, 9.343037526212950437127966657194, 9.524658547029998057303092263148, 10.45799274715920394230507479839, 10.82972934044981162317480867543, 11.43032821121379992481253538533, 11.70674144548439774937893352603, 12.33205745521536393189058579849
