| L(s) = 1 | − 8·4-s − 68·7-s + 64·16-s − 142·25-s + 544·28-s − 140·31-s + 2.78e3·49-s − 512·64-s − 644·73-s + 2.74e3·79-s − 1.14e3·97-s + 1.13e3·100-s − 3.16e3·103-s − 4.35e3·112-s + 2.63e3·121-s + 1.12e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s − 3.67·7-s + 16-s − 1.13·25-s + 3.67·28-s − 0.811·31-s + 8.11·49-s − 64-s − 1.03·73-s + 3.90·79-s − 1.20·97-s + 1.13·100-s − 3.02·103-s − 3.67·112-s + 1.97·121-s + 0.811·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3294475740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3294475740\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 142 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2630 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 1150 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38446 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 103430 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 322 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1370 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 363274 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 574 T + p^{3} T^{2} )^{2} \) |
| 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
−14.17718607903619856227228895598, −13.61939056226155163799892746589, −13.24653749272695992031803724784, −12.97062261434933240882618161972, −12.25519550759253519001320441376, −12.16288556300628572989845008256, −10.85506024077384268326097553077, −10.19958919822051231880958415977, −9.807111819011910094861810777203, −9.237740399249454937352647202806, −9.177395630532827953833012070991, −8.083141718472667870314990299212, −7.20642234479338213022816148277, −6.54035295236445653841503674898, −6.04869878860156343370791160501, −5.37971887021428514516203029817, −3.95790739940280129625087322754, −3.58957451059288321917700364229, −2.77043211972590253310789402059, −0.36614719445429008901171108053,
0.36614719445429008901171108053, 2.77043211972590253310789402059, 3.58957451059288321917700364229, 3.95790739940280129625087322754, 5.37971887021428514516203029817, 6.04869878860156343370791160501, 6.54035295236445653841503674898, 7.20642234479338213022816148277, 8.083141718472667870314990299212, 9.177395630532827953833012070991, 9.237740399249454937352647202806, 9.807111819011910094861810777203, 10.19958919822051231880958415977, 10.85506024077384268326097553077, 12.16288556300628572989845008256, 12.25519550759253519001320441376, 12.97062261434933240882618161972, 13.24653749272695992031803724784, 13.61939056226155163799892746589, 14.17718607903619856227228895598
