| L(s) = 1 | − 14·3-s + 42·5-s + 686·7-s − 698·9-s + 7.42e3·11-s + 1.18e4·13-s − 588·15-s + 1.57e4·17-s + 2.66e4·19-s − 9.60e3·21-s + 3.26e4·23-s − 1.23e5·25-s − 8.33e3·27-s − 1.58e5·29-s − 1.80e5·31-s − 1.03e5·33-s + 2.88e4·35-s − 4.58e4·37-s − 1.65e5·39-s − 3.21e5·41-s + 1.02e6·43-s − 2.93e4·45-s + 1.66e6·47-s + 3.52e5·49-s − 2.21e5·51-s − 4.10e5·53-s + 3.11e5·55-s + ⋯ |
| L(s) = 1 | − 0.299·3-s + 0.150·5-s + 0.755·7-s − 0.319·9-s + 1.68·11-s + 1.49·13-s − 0.0449·15-s + 0.779·17-s + 0.890·19-s − 0.226·21-s + 0.559·23-s − 1.57·25-s − 0.0814·27-s − 1.20·29-s − 1.08·31-s − 0.503·33-s + 0.113·35-s − 0.148·37-s − 0.447·39-s − 0.729·41-s + 1.96·43-s − 0.0479·45-s + 2.34·47-s + 3/7·49-s − 0.233·51-s − 0.378·53-s + 0.252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.757244925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.757244925\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 14 T + 298 p T^{2} + 14 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 42 T + 24986 p T^{2} - 42 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7428 T + 46542982 T^{2} - 7428 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 70 p^{2} T + 160198410 T^{2} - 70 p^{9} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 15792 T + 526157566 T^{2} - 15792 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 26614 T + 1962247086 T^{2} - 26614 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 32640 T - 991808882 T^{2} - 32640 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 158016 T + 39988772806 T^{2} + 158016 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 180740 T + 38484320958 T^{2} + 180740 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 45824 T - 18085535274 T^{2} + 45824 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 321720 T + 181439440606 T^{2} + 321720 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1023868 T + 671194246566 T^{2} - 1023868 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1665972 T + 1675477834078 T^{2} - 1665972 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 410628 T + 2334205080574 T^{2} + 410628 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1702134 T + 4026188129518 T^{2} - 1702134 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 547526 T + 5609323300002 T^{2} + 547526 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2590616 T + 4654840470246 T^{2} + 2590616 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4129272 T + 22218672158062 T^{2} - 4129272 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8008868 T + 520866105078 p T^{2} + 8008868 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2470456 T - 13654752819234 T^{2} - 2470456 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9900786 T + 68835957963214 T^{2} + 9900786 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15423492 T + 143317325773078 T^{2} - 15423492 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17377472 T + 164552259333822 T^{2} + 17377472 p^{7} T^{3} + p^{14} T^{4} \) |
| 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
−15.92452442851786385857106676770, −15.42109138499016762990100591992, −14.56400510428925775983768184847, −14.14500261212495231940646383532, −13.66650631025782096548867400247, −12.82254298316349620197232123431, −11.93850083991008694361766843338, −11.49644384530215454587305652872, −11.07652909017344734539039550225, −10.17678349432128532339976530842, −9.096038049657601570719226328952, −8.962201325185025528391773777486, −7.76717219542702270248161217236, −7.14169875331353586097768383755, −5.77783527037691559786222184223, −5.76430981346213732480806593773, −4.16526462610390347184493289631, −3.51915384741544475741790919463, −1.76456067451754075735137880176, −0.945840617285120593439029079535,
0.945840617285120593439029079535, 1.76456067451754075735137880176, 3.51915384741544475741790919463, 4.16526462610390347184493289631, 5.76430981346213732480806593773, 5.77783527037691559786222184223, 7.14169875331353586097768383755, 7.76717219542702270248161217236, 8.962201325185025528391773777486, 9.096038049657601570719226328952, 10.17678349432128532339976530842, 11.07652909017344734539039550225, 11.49644384530215454587305652872, 11.93850083991008694361766843338, 12.82254298316349620197232123431, 13.66650631025782096548867400247, 14.14500261212495231940646383532, 14.56400510428925775983768184847, 15.42109138499016762990100591992, 15.92452442851786385857106676770
