| L(s) = 1 | − 2-s − 4-s + 3·8-s − 6·9-s − 12·11-s − 16-s − 4·17-s + 6·18-s − 4·19-s + 12·22-s + 25-s − 5·32-s + 4·34-s + 6·36-s + 4·38-s + 4·41-s + 16·43-s + 12·44-s − 10·49-s − 50-s − 16·59-s + 7·64-s + 4·67-s + 4·68-s − 18·72-s − 12·73-s + 4·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 2·9-s − 3.61·11-s − 1/4·16-s − 0.970·17-s + 1.41·18-s − 0.917·19-s + 2.55·22-s + 1/5·25-s − 0.883·32-s + 0.685·34-s + 36-s + 0.648·38-s + 0.624·41-s + 2.43·43-s + 1.80·44-s − 1.42·49-s − 0.141·50-s − 2.08·59-s + 7/8·64-s + 0.488·67-s + 0.485·68-s − 2.12·72-s − 1.40·73-s + 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.69811968026233144209571661884, −7.64947192507391358342580089136, −7.21268730267367807174456588951, −6.18940471549529813746516486249, −6.01956800466047829467852819351, −5.52953758985945615238377066600, −5.11545930089817667279212368297, −4.64971827708119930945851978376, −4.39085556746158818377386484016, −3.29286190462535447792903746630, −2.95203182599403093553876063934, −2.38748749852525982096546382658, −2.08585145544051432773200925313, −0.51632190576688646302060522898, 0,
0.51632190576688646302060522898, 2.08585145544051432773200925313, 2.38748749852525982096546382658, 2.95203182599403093553876063934, 3.29286190462535447792903746630, 4.39085556746158818377386484016, 4.64971827708119930945851978376, 5.11545930089817667279212368297, 5.52953758985945615238377066600, 6.01956800466047829467852819351, 6.18940471549529813746516486249, 7.21268730267367807174456588951, 7.64947192507391358342580089136, 7.69811968026233144209571661884
