| L(s) = 1 | − 2·4-s + 13-s + 3·16-s − 2·25-s − 2·43-s − 49-s − 2·52-s − 4·61-s − 4·64-s + 2·79-s + 4·100-s + 4·103-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·4-s + 13-s + 3·16-s − 2·25-s − 2·43-s − 49-s − 2·52-s − 4·61-s − 4·64-s + 2·79-s + 4·100-s + 4·103-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9979281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9979281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5402350802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5402350802\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−8.970454301840378114844270852522, −8.675352208516013690345508136431, −8.474290167910083485739218860985, −7.85873949753974626418585793437, −7.74589462278682328370679965733, −7.51338572946554106529909414535, −6.66648023416483350188419857829, −6.16036136808540049729376736211, −6.08977932665438184566518291981, −5.63860295277060159178014739451, −5.03092569191006561797037963706, −4.80591425104734348153857136979, −4.52236192596008570390918028170, −3.88309409147279064740397030240, −3.54627421091715730801558074232, −3.42267278611860647736783694640, −2.70533922214901009490876624324, −1.65813500017098320134657815927, −1.55703875802624110510365532468, −0.48261649647040078028399855955,
0.48261649647040078028399855955, 1.55703875802624110510365532468, 1.65813500017098320134657815927, 2.70533922214901009490876624324, 3.42267278611860647736783694640, 3.54627421091715730801558074232, 3.88309409147279064740397030240, 4.52236192596008570390918028170, 4.80591425104734348153857136979, 5.03092569191006561797037963706, 5.63860295277060159178014739451, 6.08977932665438184566518291981, 6.16036136808540049729376736211, 6.66648023416483350188419857829, 7.51338572946554106529909414535, 7.74589462278682328370679965733, 7.85873949753974626418585793437, 8.474290167910083485739218860985, 8.675352208516013690345508136431, 8.970454301840378114844270852522
