| L(s) = 1 | + 2-s + 4-s + 3·5-s − 4·7-s + 8-s − 2·9-s + 3·10-s − 4·14-s + 16-s + 6·17-s − 2·18-s − 2·19-s + 3·20-s + 4·25-s − 4·28-s + 32-s + 6·34-s − 12·35-s − 2·36-s + 14·37-s − 2·38-s + 3·40-s − 6·41-s + 8·43-s − 6·45-s + 7·49-s + 4·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.471·18-s − 0.458·19-s + 0.670·20-s + 4/5·25-s − 0.755·28-s + 0.176·32-s + 1.02·34-s − 2.02·35-s − 1/3·36-s + 2.30·37-s − 0.324·38-s + 0.474·40-s − 0.937·41-s + 1.21·43-s − 0.894·45-s + 49-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 744200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 744200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.313436071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.313436071\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 61 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.273103582590453710447369110356, −7.73415358506455885950976473725, −7.32375523179638897099752127496, −6.69997914813041705577253535087, −6.30911355705745093879371726578, −5.97365518916299418590409607077, −5.71906710358726880267334569647, −5.27504029482227875120645671074, −4.58418352369754639460993736805, −4.02748106314744709295042452782, −3.32053268697952295295718457450, −2.98124788655601182296103151998, −2.50288753439762318151560938438, −1.79452254735806948407533377767, −0.788954951867716415293555410825,
0.788954951867716415293555410825, 1.79452254735806948407533377767, 2.50288753439762318151560938438, 2.98124788655601182296103151998, 3.32053268697952295295718457450, 4.02748106314744709295042452782, 4.58418352369754639460993736805, 5.27504029482227875120645671074, 5.71906710358726880267334569647, 5.97365518916299418590409607077, 6.30911355705745093879371726578, 6.69997914813041705577253535087, 7.32375523179638897099752127496, 7.73415358506455885950976473725, 8.273103582590453710447369110356
