L(s) = 1 | + 7-s + 12·11-s + 12·23-s − 10·25-s − 12·29-s + 4·37-s − 8·43-s + 49-s + 12·53-s + 16·67-s − 12·71-s + 12·77-s − 8·79-s + 12·107-s + 28·109-s + 12·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 12·161-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 3.61·11-s + 2.50·23-s − 2·25-s − 2.22·29-s + 0.657·37-s − 1.21·43-s + 1/7·49-s + 1.64·53-s + 1.95·67-s − 1.42·71-s + 1.36·77-s − 0.900·79-s + 1.16·107-s + 2.68·109-s + 1.12·113-s + 7.81·121-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 + 0.945·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.637985935\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.637985935\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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
−8.729935221736740713914818138949, −8.300043091642042619777544931378, −7.44499063873378776873294009662, −7.12758834797060137294534233139, −6.93446262615281462668117301613, −6.11056271857714724112691707067, −6.03492627650465326217694729948, −5.32133833421984252130695309604, −4.63917458628061830491997962672, −4.15927538579926872458234038883, −3.53462971911322619523210724383, −3.52771007221053761224054171247, −2.20338451459941845252791108478, −1.58867103529813382887688476545, −1.00219484121657416616236254344,
1.00219484121657416616236254344, 1.58867103529813382887688476545, 2.20338451459941845252791108478, 3.52771007221053761224054171247, 3.53462971911322619523210724383, 4.15927538579926872458234038883, 4.63917458628061830491997962672, 5.32133833421984252130695309604, 6.03492627650465326217694729948, 6.11056271857714724112691707067, 6.93446262615281462668117301613, 7.12758834797060137294534233139, 7.44499063873378776873294009662, 8.300043091642042619777544931378, 8.729935221736740713914818138949