| L(s) = 1 | − 6·4-s − 30·7-s − 70·13-s − 28·16-s + 182·19-s + 180·28-s − 294·31-s − 740·37-s − 670·43-s − 11·49-s + 420·52-s + 854·61-s + 552·64-s − 30·67-s + 140·73-s − 1.09e3·76-s − 1.75e3·79-s + 2.10e3·91-s + 2.17e3·97-s − 3.08e3·103-s − 1.62e3·109-s + 840·112-s + 1.33e3·121-s + 1.76e3·124-s + 127-s + 131-s − 5.46e3·133-s + ⋯ |
| L(s) = 1 | − 3/4·4-s − 1.61·7-s − 1.49·13-s − 0.437·16-s + 2.19·19-s + 1.21·28-s − 1.70·31-s − 3.28·37-s − 2.37·43-s − 0.0320·49-s + 1.12·52-s + 1.79·61-s + 1.07·64-s − 0.0547·67-s + 0.224·73-s − 1.64·76-s − 2.49·79-s + 2.41·91-s + 2.27·97-s − 2.94·103-s − 1.42·109-s + 0.708·112-s + 1.00·121-s + 1.27·124-s + 0.000698·127-s + 0.000666·131-s − 3.55·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 p T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1338 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 35 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1986 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 91 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 11374 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 44778 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 147 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 p T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 58158 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 335 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 176286 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 289914 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 373242 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 p T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 711822 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 876 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 861334 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1085 T + p^{3} T^{2} )^{2} \) |
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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
−11.72308143352839855360762285693, −11.15314918098509793744635311909, −10.24955774852636271393972652887, −10.01749505805454377297480946277, −9.629068903512475056544013502066, −9.257855734092674488226791836294, −8.746893228805357428096158941487, −8.165420628785798666656475781683, −7.23311097739319602963219167242, −7.09264583882015994401186527988, −6.65035031234585382071055308059, −5.66571065665985945958976135147, −5.13504617309783731748527365841, −4.91601179539327128908461726485, −3.61392382118507272374485055159, −3.51576106426752832375520331175, −2.67610166586162000289489916545, −1.59369093416360635764731345696, 0, 0,
1.59369093416360635764731345696, 2.67610166586162000289489916545, 3.51576106426752832375520331175, 3.61392382118507272374485055159, 4.91601179539327128908461726485, 5.13504617309783731748527365841, 5.66571065665985945958976135147, 6.65035031234585382071055308059, 7.09264583882015994401186527988, 7.23311097739319602963219167242, 8.165420628785798666656475781683, 8.746893228805357428096158941487, 9.257855734092674488226791836294, 9.629068903512475056544013502066, 10.01749505805454377297480946277, 10.24955774852636271393972652887, 11.15314918098509793744635311909, 11.72308143352839855360762285693
