| L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s + 8-s − 2·9-s − 3·10-s − 12-s + 3·15-s + 16-s − 2·18-s − 3·20-s + 3·23-s − 24-s + 3·25-s + 5·27-s − 15·29-s + 3·30-s + 32-s − 2·36-s − 3·40-s − 7·43-s + 6·45-s + 3·46-s − 9·47-s − 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.948·10-s − 0.288·12-s + 0.774·15-s + 1/4·16-s − 0.471·18-s − 0.670·20-s + 0.625·23-s − 0.204·24-s + 3/5·25-s + 0.962·27-s − 2.78·29-s + 0.547·30-s + 0.176·32-s − 1/3·36-s − 0.474·40-s − 1.06·43-s + 0.894·45-s + 0.442·46-s − 1.31·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−9.840742682098522820063066756252, −9.132664673020184816660665585105, −8.742920387793120833329669211237, −7.992165726659959850244785398515, −7.63396913798533183281661803963, −7.19514499220587777180573770192, −6.44941669893118025902969792162, −6.01701451858761095209542034312, −5.29995722454610253310903341779, −4.87201918137276518516228293607, −4.18393970924083941624739799108, −3.46150685878293447687979124640, −3.07647253978264066773470418655, −1.78700892975528860038071577592, 0,
1.78700892975528860038071577592, 3.07647253978264066773470418655, 3.46150685878293447687979124640, 4.18393970924083941624739799108, 4.87201918137276518516228293607, 5.29995722454610253310903341779, 6.01701451858761095209542034312, 6.44941669893118025902969792162, 7.19514499220587777180573770192, 7.63396913798533183281661803963, 7.992165726659959850244785398515, 8.742920387793120833329669211237, 9.132664673020184816660665585105, 9.840742682098522820063066756252
