| L(s) = 1 | − 64·7-s + 196·17-s − 64·23-s + 106·25-s − 512·31-s + 204·41-s + 640·47-s + 2.38e3·49-s + 832·71-s − 276·73-s − 128·79-s − 1.16e3·89-s + 476·97-s + 1.98e3·103-s + 604·113-s − 1.25e4·119-s + 2.59e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4.09e3·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.45·7-s + 2.79·17-s − 0.580·23-s + 0.847·25-s − 2.96·31-s + 0.777·41-s + 1.98·47-s + 6.95·49-s + 1.39·71-s − 0.442·73-s − 0.182·79-s − 1.38·89-s + 0.498·97-s + 1.89·103-s + 0.502·113-s − 9.66·119-s + 1.95·121-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 + 2.00·161-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.830119929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.830119929\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 106 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2598 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 98 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 5974 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 19194 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 256 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 92842 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 71398 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 320 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 291978 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 244294 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 49466 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 296822 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 416 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 138 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 64 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 989910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 582 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 238 T + p^{3} T^{2} )^{2} \) |
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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
−9.666664658809018361517647869113, −9.392656189466090410047124048107, −8.890809329735441799240492357002, −8.674862319785661612836621078538, −7.66418723613323904447223764772, −7.58914452445000718275572054090, −7.07104639519245952422796475666, −6.79746673901580801844728352654, −6.15714614460348207807664741314, −5.85967078359538650728688935205, −5.60699025300210885792316331158, −5.17957456048342191021504473275, −4.00312819470748420122643950859, −3.88892057072972161905692027248, −3.25915613963670719844321856028, −3.15367373787743586084644408835, −2.59077756564567735159746202525, −1.74096922812398386298285286014, −0.67341887266093627643859601061, −0.52020198516688606631277704698,
0.52020198516688606631277704698, 0.67341887266093627643859601061, 1.74096922812398386298285286014, 2.59077756564567735159746202525, 3.15367373787743586084644408835, 3.25915613963670719844321856028, 3.88892057072972161905692027248, 4.00312819470748420122643950859, 5.17957456048342191021504473275, 5.60699025300210885792316331158, 5.85967078359538650728688935205, 6.15714614460348207807664741314, 6.79746673901580801844728352654, 7.07104639519245952422796475666, 7.58914452445000718275572054090, 7.66418723613323904447223764772, 8.674862319785661612836621078538, 8.890809329735441799240492357002, 9.392656189466090410047124048107, 9.666664658809018361517647869113
