| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s − 6·9-s + 2·10-s + 4·13-s − 16-s − 4·17-s + 6·18-s + 2·20-s + 3·25-s − 4·26-s − 2·29-s − 5·32-s + 4·34-s + 6·36-s + 20·37-s − 6·40-s + 4·41-s + 12·45-s − 10·49-s − 3·50-s − 4·52-s − 12·53-s + 2·58-s − 12·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s − 2·9-s + 0.632·10-s + 1.10·13-s − 1/4·16-s − 0.970·17-s + 1.41·18-s + 0.447·20-s + 3/5·25-s − 0.784·26-s − 0.371·29-s − 0.883·32-s + 0.685·34-s + 36-s + 3.28·37-s − 0.948·40-s + 0.624·41-s + 1.78·45-s − 1.42·49-s − 0.424·50-s − 0.554·52-s − 1.64·53-s + 0.262·58-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.402368279712544431210177649490, −8.257954938132273132829266439629, −7.64947192507391358342580089136, −7.59896035728415497254929554682, −6.55138204331491112705378355339, −6.01956800466047829467852819351, −6.00486449332409074941837987010, −4.95898829319521499302586493249, −4.64971827708119930945851978376, −4.08126465874364407576578580037, −3.38444605735957405294650498855, −2.95203182599403093553876063934, −2.08605131316588777212252155975, −0.921963524041875158976208258286, 0,
0.921963524041875158976208258286, 2.08605131316588777212252155975, 2.95203182599403093553876063934, 3.38444605735957405294650498855, 4.08126465874364407576578580037, 4.64971827708119930945851978376, 4.95898829319521499302586493249, 6.00486449332409074941837987010, 6.01956800466047829467852819351, 6.55138204331491112705378355339, 7.59896035728415497254929554682, 7.64947192507391358342580089136, 8.257954938132273132829266439629, 8.402368279712544431210177649490
