| L(s) = 1 | + 3·2-s + 5·4-s + 33·5-s + 99·8-s + 99·10-s + 1.13e3·11-s + 925·13-s − 459·16-s + 324·17-s + 2.31e3·19-s + 165·20-s + 3.41e3·22-s − 1.59e3·23-s − 2.38e3·25-s + 2.77e3·26-s + 2.21e3·29-s + 4.29e3·31-s − 4.36e3·32-s + 972·34-s − 1.91e4·37-s + 6.93e3·38-s + 3.26e3·40-s + 1.28e4·41-s − 2.77e3·43-s + 5.68e3·44-s − 4.78e3·46-s + 2.31e4·47-s + ⋯ |
| L(s) = 1 | + 0.530·2-s + 5/32·4-s + 0.590·5-s + 0.546·8-s + 0.313·10-s + 2.83·11-s + 1.51·13-s − 0.448·16-s + 0.271·17-s + 1.46·19-s + 0.0922·20-s + 1.50·22-s − 0.629·23-s − 0.762·25-s + 0.805·26-s + 0.489·29-s + 0.802·31-s − 0.753·32-s + 0.144·34-s − 2.29·37-s + 0.778·38-s + 0.322·40-s + 1.19·41-s − 0.228·43-s + 0.442·44-s − 0.333·46-s + 1.52·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.056862091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.056862091\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + p^{2} T^{2} - 3 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 33 T + 3472 T^{2} - 33 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 1137 T + 645232 T^{2} - 1137 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 925 T + 951450 T^{2} - 925 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 324 T + 1502434 T^{2} - 324 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2311 T + 6241998 T^{2} - 2311 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1596 T + 7604206 T^{2} + 1596 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2217 T + 42125014 T^{2} - 2217 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4294 T + 25151367 T^{2} - 4294 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 19109 T + 184470078 T^{2} + 19109 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12858 T + 228491122 T^{2} - 12858 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2771 T + 36114396 T^{2} + 2771 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 23160 T + 570076618 T^{2} - 23160 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 31653 T + 896010526 T^{2} - 31653 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 41097 T + 896994610 T^{2} + 41097 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 42052 T + 1728407262 T^{2} - 42052 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 30763 T + 1698033324 T^{2} - 30763 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 102096 T + 6091648810 T^{2} + 102096 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 28577 T + 2636499108 T^{2} + 28577 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18464 T + 3332046261 T^{2} + 18464 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 61179 T + 8589312784 T^{2} + 61179 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 29322 T + 11382011794 T^{2} + 29322 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9791 T + 17134261944 T^{2} + 9791 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.48688810960663029427283574015, −10.15670163445372126028916953913, −9.539026542696432615660624280125, −9.235606943084840546019348601942, −8.684850933300064792974571476858, −8.573334065297954853048648904935, −7.67117074420058788260545161249, −7.02596325487524799379895398119, −6.89968542849825131146364519967, −6.24497820982057758249024144696, −5.71055750718579183677579920587, −5.63761477489905406755970060156, −4.48015639875126506976308471848, −4.30506551096447963731067089022, −3.60120163763069927001350115399, −3.43162232135470607513512599127, −2.36918199094573995683757906077, −1.54558195139549606126964651959, −1.36289694726182339144982468708, −0.70696654682940644707459410538,
0.70696654682940644707459410538, 1.36289694726182339144982468708, 1.54558195139549606126964651959, 2.36918199094573995683757906077, 3.43162232135470607513512599127, 3.60120163763069927001350115399, 4.30506551096447963731067089022, 4.48015639875126506976308471848, 5.63761477489905406755970060156, 5.71055750718579183677579920587, 6.24497820982057758249024144696, 6.89968542849825131146364519967, 7.02596325487524799379895398119, 7.67117074420058788260545161249, 8.573334065297954853048648904935, 8.684850933300064792974571476858, 9.235606943084840546019348601942, 9.539026542696432615660624280125, 10.15670163445372126028916953913, 10.48688810960663029427283574015
