| L(s) = 1 | − 9·9-s − 44·19-s − 4·31-s − 98·49-s − 236·61-s + 196·79-s + 81·81-s + 44·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 338·169-s + 396·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s − 2.31·19-s − 0.129·31-s − 2·49-s − 3.86·61-s + 2.48·79-s + 81-s + 0.403·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2·169-s + 2.31·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2523396029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2523396029\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 382 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 98 T^{2} + p^{4} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4222 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1778 T^{2} + p^{4} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 118 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 98 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9938 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
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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
−9.579636158406318047917841490552, −9.358092223404747095507577888221, −8.902953340774729244888414740141, −8.591859311204909485854456928467, −8.021402629648115208490292992016, −7.956787023903309235597167625612, −7.35600749831495715759144378672, −6.63826214934405337576464961003, −6.48293632829349111989369110147, −5.99276270125465020971921798772, −5.72900622309570724465978641752, −4.87882451913006771059841710368, −4.71604365003280570270281432213, −4.20691802808942260616970166799, −3.41379623888030203423208867701, −3.23092271652236388101507465236, −2.37638601668693556262988591472, −2.06061834793225165097105817816, −1.27393243802652374639847685657, −0.14864680354485370019216411565,
0.14864680354485370019216411565, 1.27393243802652374639847685657, 2.06061834793225165097105817816, 2.37638601668693556262988591472, 3.23092271652236388101507465236, 3.41379623888030203423208867701, 4.20691802808942260616970166799, 4.71604365003280570270281432213, 4.87882451913006771059841710368, 5.72900622309570724465978641752, 5.99276270125465020971921798772, 6.48293632829349111989369110147, 6.63826214934405337576464961003, 7.35600749831495715759144378672, 7.956787023903309235597167625612, 8.021402629648115208490292992016, 8.591859311204909485854456928467, 8.902953340774729244888414740141, 9.358092223404747095507577888221, 9.579636158406318047917841490552
