| L(s) = 1 | + 8·13-s − 8·25-s + 4·37-s + 14·49-s − 20·61-s + 32·73-s − 16·97-s − 40·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.21·13-s − 8/5·25-s + 0.657·37-s + 2·49-s − 2.56·61-s + 3.74·73-s − 1.62·97-s − 3.83·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.231311325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.231311325\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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
−13.66087435228723288266073593815, −12.94860897294875244855465314339, −12.16288692014256542391037416544, −12.01926027725275422018720232818, −11.06515689393387037656719208575, −11.01743364068965368102355449919, −10.45561598678540356209631569305, −9.656087274818344088634370141675, −9.256047475851871712058476339581, −8.682770101656673594743778626989, −8.047594690926185793623914759732, −7.74240346995427529361484942778, −6.82380403811947813379485228639, −6.20960679757941866437092235792, −5.83724029182699751068136356638, −5.08339899657969180788829960218, −3.91618276927594421014409614953, −3.82882281227794167043136970311, −2.59850145117782529263796106565, −1.37578877750920465414474926871,
1.37578877750920465414474926871, 2.59850145117782529263796106565, 3.82882281227794167043136970311, 3.91618276927594421014409614953, 5.08339899657969180788829960218, 5.83724029182699751068136356638, 6.20960679757941866437092235792, 6.82380403811947813379485228639, 7.74240346995427529361484942778, 8.047594690926185793623914759732, 8.682770101656673594743778626989, 9.256047475851871712058476339581, 9.656087274818344088634370141675, 10.45561598678540356209631569305, 11.01743364068965368102355449919, 11.06515689393387037656719208575, 12.01926027725275422018720232818, 12.16288692014256542391037416544, 12.94860897294875244855465314339, 13.66087435228723288266073593815
