| L(s) = 1 | + 5·2-s + 20·3-s + 15·4-s + 100·6-s + 200·7-s + 185·8-s + 55·9-s − 196·11-s + 300·12-s − 360·13-s + 1.00e3·14-s + 551·16-s + 1.49e3·17-s + 275·18-s − 3.18e3·19-s + 4.00e3·21-s − 980·22-s + 1.56e3·23-s + 3.70e3·24-s − 1.80e3·26-s − 940·27-s + 3.00e3·28-s − 3.92e3·29-s − 1.09e3·31-s − 1.81e3·32-s − 3.92e3·33-s + 7.45e3·34-s + ⋯ |
| L(s) = 1 | + 0.883·2-s + 1.28·3-s + 0.468·4-s + 1.13·6-s + 1.54·7-s + 1.02·8-s + 0.226·9-s − 0.488·11-s + 0.601·12-s − 0.590·13-s + 1.36·14-s + 0.538·16-s + 1.25·17-s + 0.200·18-s − 2.02·19-s + 1.97·21-s − 0.431·22-s + 0.614·23-s + 1.31·24-s − 0.522·26-s − 0.248·27-s + 0.723·28-s − 0.865·29-s − 0.204·31-s − 0.313·32-s − 0.626·33-s + 1.10·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.955558728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.955558728\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 5 T + 5 p T^{2} - 5 p^{5} T^{3} + p^{10} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 20 T + 115 p T^{2} - 20 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 200 T + 42650 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 196 T + 181081 T^{2} + 196 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 360 T + 713290 T^{2} + 360 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1490 T + 2280355 T^{2} - 1490 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3180 T + 7185073 T^{2} + 3180 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1560 T + 13472410 T^{2} - 1560 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3920 T + 44478298 T^{2} + 3920 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 1096 T - 15343894 T^{2} + 1096 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2020 T + 130823790 T^{2} - 2020 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 27754 T + 414643531 T^{2} - 27754 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3000 T + 23431750 T^{2} + 3000 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 25760 T + 480952270 T^{2} - 25760 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 26980 T + 984299470 T^{2} + 26980 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11960 T + 1234152598 T^{2} - 11960 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 24396 T + 1596983806 T^{2} + 24396 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 40060 T + 2949898505 T^{2} + 40060 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 87296 T + 5453356606 T^{2} + 87296 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 70290 T + 5306812435 T^{2} + 70290 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 65480 T + 5707696298 T^{2} - 65480 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 92580 T + 9976491505 T^{2} - 92580 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 72810 T + 5926578523 T^{2} + 72810 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 126140 T + 13936294470 T^{2} + 126140 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−16.76782391118179749278291581339, −16.16194588942137844406835331320, −14.99655330484950880911238714015, −14.89506302804177742410775015407, −14.25298072430603815614484980075, −14.09117512307729399236810266876, −13.06354839012604626163844270933, −12.75011724062132026492557319314, −11.77133449111415275199220184935, −10.87458170338699503408965297617, −10.58993952985139962514719176689, −9.296757616601543303901014697057, −8.572227182224539262010358264535, −7.74365534724017738266669314159, −7.49507376360073757576428114929, −5.89181490410979452593350104783, −4.84461700158168597309980642907, −4.18882828830997964646998933624, −2.78687560275575044330968615714, −1.79831899151529647478138098756,
1.79831899151529647478138098756, 2.78687560275575044330968615714, 4.18882828830997964646998933624, 4.84461700158168597309980642907, 5.89181490410979452593350104783, 7.49507376360073757576428114929, 7.74365534724017738266669314159, 8.572227182224539262010358264535, 9.296757616601543303901014697057, 10.58993952985139962514719176689, 10.87458170338699503408965297617, 11.77133449111415275199220184935, 12.75011724062132026492557319314, 13.06354839012604626163844270933, 14.09117512307729399236810266876, 14.25298072430603815614484980075, 14.89506302804177742410775015407, 14.99655330484950880911238714015, 16.16194588942137844406835331320, 16.76782391118179749278291581339
