| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s + 13-s + 16-s + 18-s − 20-s − 4·25-s + 26-s − 7·29-s + 32-s + 36-s − 37-s − 40-s + 6·41-s − 45-s − 5·49-s − 4·50-s + 52-s − 8·53-s − 7·58-s + 5·61-s + 64-s − 65-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.277·13-s + 1/4·16-s + 0.235·18-s − 0.223·20-s − 4/5·25-s + 0.196·26-s − 1.29·29-s + 0.176·32-s + 1/6·36-s − 0.164·37-s − 0.158·40-s + 0.937·41-s − 0.149·45-s − 5/7·49-s − 0.565·50-s + 0.138·52-s − 1.09·53-s − 0.919·58-s + 0.640·61-s + 1/8·64-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.66804818133771859555476774633, −7.40355979655295743234939295094, −6.94285053630716929145923761321, −6.39877039152888960288289665850, −6.04184257205408028744128499258, −5.52509668961788763808008711965, −5.18400474440040465338540903737, −4.48411646003675519588745879317, −4.15953254495659941842151980186, −3.73060513267072016621174577960, −3.19034800470271593143771104117, −2.61357363277461712846398766373, −1.87975513770260881427409019987, −1.29455181987133681133254273065, 0,
1.29455181987133681133254273065, 1.87975513770260881427409019987, 2.61357363277461712846398766373, 3.19034800470271593143771104117, 3.73060513267072016621174577960, 4.15953254495659941842151980186, 4.48411646003675519588745879317, 5.18400474440040465338540903737, 5.52509668961788763808008711965, 6.04184257205408028744128499258, 6.39877039152888960288289665850, 6.94285053630716929145923761321, 7.40355979655295743234939295094, 7.66804818133771859555476774633
