| L(s) = 1 | − 32·2-s + 512·4-s + 1.43e3·5-s − 4.59e4·10-s − 2.85e5·13-s − 2.62e5·16-s − 9.62e5·17-s + 7.35e5·20-s + 1.08e5·25-s + 9.12e6·26-s + 8.38e6·32-s + 3.08e7·34-s − 2.46e7·37-s + 1.51e7·41-s − 3.48e6·50-s − 1.45e8·52-s − 2.40e7·53-s + 4.32e8·61-s − 1.34e8·64-s − 4.09e8·65-s − 4.92e8·68-s + 5.25e8·73-s + 7.88e8·74-s − 3.76e8·80-s − 3.87e8·81-s − 4.83e8·82-s − 1.38e9·85-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.02·5-s − 1.45·10-s − 2.76·13-s − 16-s − 2.79·17-s + 1.02·20-s + 0.0557·25-s + 3.91·26-s + 1.41·32-s + 3.95·34-s − 2.16·37-s + 0.835·41-s − 0.0789·50-s − 2.76·52-s − 0.418·53-s + 3.99·61-s − 64-s − 2.84·65-s − 2.79·68-s + 2.16·73-s + 3.05·74-s − 1.02·80-s − 81-s − 1.18·82-s − 2.87·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.4300272283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4300272283\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( 1 + p^{5} T + p^{9} T^{2} \) |
| 5 | $C_2$ | \( 1 - 1436 T + p^{9} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{18} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 112806 T + p^{9} T^{2} )( 1 + 172316 T + p^{9} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 407992 T + p^{9} T^{2} )( 1 + 554882 T + p^{9} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7314710 T + p^{9} T^{2} )( 1 + 7314710 T + p^{9} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1923372 T + p^{9} T^{2} )( 1 + 22718882 T + p^{9} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 7561912 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{18} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{18} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 68323684 T + p^{9} T^{2} )( 1 + 92363026 T + p^{9} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 216178092 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 483419504 T + p^{9} T^{2} )( 1 - 42331194 T + p^{9} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1125568310 T + p^{9} T^{2} )( 1 + 1125568310 T + p^{9} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 1016663992 T + p^{9} T^{2} )( 1 + 1416798702 T + p^{9} T^{2} ) \) |
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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
−17.10834291643668960401460599236, −15.94682765643763787992202291048, −15.48633157619524382335517816725, −14.58774323997186647190519514962, −13.95521152903699283986481302083, −13.20833954016256245732515046094, −12.52754040406852759140574793664, −11.57151564675709861886062856319, −10.83580556494504077604005262131, −10.04794477717371689693507769837, −9.615891131265515284053788425339, −9.004134441672658958668915891532, −8.216141406435621666441702190787, −6.97908946006851454030779892842, −6.86941049264641975970565320729, −5.29454546357434189358783036787, −4.47942169225095286548021910915, −2.20090002811024740047050399195, −2.19057452331915443735846917327, −0.36121756008191639186892925609,
0.36121756008191639186892925609, 2.19057452331915443735846917327, 2.20090002811024740047050399195, 4.47942169225095286548021910915, 5.29454546357434189358783036787, 6.86941049264641975970565320729, 6.97908946006851454030779892842, 8.216141406435621666441702190787, 9.004134441672658958668915891532, 9.615891131265515284053788425339, 10.04794477717371689693507769837, 10.83580556494504077604005262131, 11.57151564675709861886062856319, 12.52754040406852759140574793664, 13.20833954016256245732515046094, 13.95521152903699283986481302083, 14.58774323997186647190519514962, 15.48633157619524382335517816725, 15.94682765643763787992202291048, 17.10834291643668960401460599236
