L(s) = 1 | + 3·4-s + 4·7-s + 10·13-s + 5·16-s − 3·25-s + 12·28-s + 8·31-s − 6·37-s + 12·43-s − 2·49-s + 30·52-s − 12·61-s + 3·64-s − 20·67-s − 20·73-s − 16·79-s + 40·91-s + 10·97-s − 9·100-s + 28·103-s + 10·109-s + 20·112-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 3/2·4-s + 1.51·7-s + 2.77·13-s + 5/4·16-s − 3/5·25-s + 2.26·28-s + 1.43·31-s − 0.986·37-s + 1.82·43-s − 2/7·49-s + 4.16·52-s − 1.53·61-s + 3/8·64-s − 2.44·67-s − 2.34·73-s − 1.80·79-s + 4.19·91-s + 1.01·97-s − 0.899·100-s + 2.75·103-s + 0.957·109-s + 1.88·112-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.909407779\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.909407779\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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
−10.42050543841446949014843697780, −9.823301014418404707615408714952, −8.888118693639745723205299374602, −8.779446533560007743599347167638, −8.569574567281581248330849944299, −7.77594574784300739537262892995, −7.66517707877848537405587208908, −7.35922998547033483473724572447, −6.55220249854625099979645502718, −6.16295820843450353391913019912, −6.08939320500831878326768482673, −5.56139704051060845600743152726, −4.87924307440615116349372274396, −4.29707997991328737318830135800, −3.99480843407406712831031412201, −3.03439835762242077521563939459, −3.01381741469607397174690710148, −1.90608658468002312089244101013, −1.59089441543721140954265628235, −1.12509458117987250833598923208,
1.12509458117987250833598923208, 1.59089441543721140954265628235, 1.90608658468002312089244101013, 3.01381741469607397174690710148, 3.03439835762242077521563939459, 3.99480843407406712831031412201, 4.29707997991328737318830135800, 4.87924307440615116349372274396, 5.56139704051060845600743152726, 6.08939320500831878326768482673, 6.16295820843450353391913019912, 6.55220249854625099979645502718, 7.35922998547033483473724572447, 7.66517707877848537405587208908, 7.77594574784300739537262892995, 8.569574567281581248330849944299, 8.779446533560007743599347167638, 8.888118693639745723205299374602, 9.823301014418404707615408714952, 10.42050543841446949014843697780