| L(s) = 1 | + 7·4-s − 9·9-s − 12·11-s − 15·16-s + 8·19-s + 246·29-s − 410·31-s − 63·36-s + 114·41-s − 84·44-s − 49·49-s − 66·59-s − 854·61-s − 553·64-s + 600·71-s + 56·76-s − 1.37e3·79-s + 81·81-s − 1.42e3·89-s + 108·99-s + 1.24e3·101-s − 376·109-s + 1.72e3·116-s − 2.55e3·121-s − 2.87e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/8·4-s − 1/3·9-s − 0.328·11-s − 0.234·16-s + 0.0965·19-s + 1.57·29-s − 2.37·31-s − 0.291·36-s + 0.434·41-s − 0.287·44-s − 1/7·49-s − 0.145·59-s − 1.79·61-s − 1.08·64-s + 1.00·71-s + 0.0845·76-s − 1.95·79-s + 1/9·81-s − 1.70·89-s + 0.109·99-s + 1.22·101-s − 0.330·109-s + 1.37·116-s − 1.91·121-s − 2.07·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.986549765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.986549765\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2713 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9097 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 18709 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 123 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 205 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 32662 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 57 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6635 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 204046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 190825 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 33 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 p T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 207142 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 300 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 768430 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 686 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 819227 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 714 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1581310 T^{2} + p^{6} T^{4} \) |
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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
−11.03068521985459467787473909131, −10.43021886634333702103642260313, −9.741850912249156030007653062573, −9.424659989844692107675341905916, −8.753710698527933397337451968172, −8.600410557921557387534390544180, −7.85830131261247754008730893827, −7.43936081452090974535221489084, −7.15127381406718483297750174762, −6.46589151938529893057921112130, −6.23023225345036466285864938668, −5.52732903390309786733061586490, −5.19005134466576327480953875710, −4.44789671676380519122996132935, −3.96539393725538213397556076459, −3.05647787097957371091734738498, −2.84169379704529774093274675873, −2.01453298339044240041731029889, −1.48460782437905934716613207419, −0.41300850095057475328406743596,
0.41300850095057475328406743596, 1.48460782437905934716613207419, 2.01453298339044240041731029889, 2.84169379704529774093274675873, 3.05647787097957371091734738498, 3.96539393725538213397556076459, 4.44789671676380519122996132935, 5.19005134466576327480953875710, 5.52732903390309786733061586490, 6.23023225345036466285864938668, 6.46589151938529893057921112130, 7.15127381406718483297750174762, 7.43936081452090974535221489084, 7.85830131261247754008730893827, 8.600410557921557387534390544180, 8.753710698527933397337451968172, 9.424659989844692107675341905916, 9.741850912249156030007653062573, 10.43021886634333702103642260313, 11.03068521985459467787473909131
