| L(s) = 1 | − 4-s − 2·9-s − 4·11-s + 16-s − 25-s + 2·36-s + 4·44-s − 2·49-s − 4·59-s − 64-s + 3·81-s − 4·89-s + 8·99-s + 100-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s − 2·9-s − 4·11-s + 16-s − 25-s + 2·36-s + 4·44-s − 2·49-s − 4·59-s − 64-s + 3·81-s − 4·89-s + 8·99-s + 100-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4494400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4494400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0009773116069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0009773116069\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 53 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$ | \( ( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.834351539218774639745079152317, −8.828608953911771686740610807995, −8.786624232844280443057044639219, −8.082976992665251332969773213839, −8.074776315276835614984628848424, −7.65726742665146163087783718337, −7.61749558780773721583379165357, −6.67773970297767010509645751551, −6.03409017643960610334596743380, −5.84586092623743720130495484098, −5.32570532465154841480986250527, −5.29066890683433213454716655069, −4.73788322706441855345798598895, −4.45771865789542805468868672500, −3.51515185690551718098050273075, −3.14204862990300261520076665603, −2.72002055196459542850928229754, −2.52793211543000979453116367514, −1.63870836448779679466973289881, −0.02079224473519950532765223900,
0.02079224473519950532765223900, 1.63870836448779679466973289881, 2.52793211543000979453116367514, 2.72002055196459542850928229754, 3.14204862990300261520076665603, 3.51515185690551718098050273075, 4.45771865789542805468868672500, 4.73788322706441855345798598895, 5.29066890683433213454716655069, 5.32570532465154841480986250527, 5.84586092623743720130495484098, 6.03409017643960610334596743380, 6.67773970297767010509645751551, 7.61749558780773721583379165357, 7.65726742665146163087783718337, 8.074776315276835614984628848424, 8.082976992665251332969773213839, 8.786624232844280443057044639219, 8.828608953911771686740610807995, 9.834351539218774639745079152317
