| L(s) = 1 | + 2·4-s − 2·9-s − 5·16-s + 12·29-s − 4·36-s + 20·49-s − 24·59-s − 20·64-s − 15·81-s − 20·109-s + 24·116-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 10·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s − 2/3·9-s − 5/4·16-s + 2.22·29-s − 2/3·36-s + 20/7·49-s − 3.12·59-s − 5/2·64-s − 5/3·81-s − 1.91·109-s + 2.22·116-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.832240217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.832240217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 89 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 113 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.51142627490353454858613080239, −7.22715189671199119391494709357, −6.91043872904680642404314832304, −6.76354037176925662676899740126, −6.60171800811418718193766658615, −6.48774372460547012649762794703, −5.90596707384285186231098178335, −5.88932420505269698424430521665, −5.73015874903168392778394292830, −5.57191954976078894604280606031, −4.90555867537011437763193333636, −4.73284474494995545118946160614, −4.58721714779760185124417103057, −4.54786764157570828730816389595, −3.89509090474010465262003198840, −3.88801436995242210653300818263, −3.35057658828239524947505774145, −3.00849050210456249869652686545, −2.83200533514661363503352640209, −2.56550750020201812081216460476, −2.28259647281828210122337385569, −2.01216404094787750783470506256, −1.40048204825684997757517577898, −1.16230939407100343270898103503, −0.35549355000695624129134292350,
0.35549355000695624129134292350, 1.16230939407100343270898103503, 1.40048204825684997757517577898, 2.01216404094787750783470506256, 2.28259647281828210122337385569, 2.56550750020201812081216460476, 2.83200533514661363503352640209, 3.00849050210456249869652686545, 3.35057658828239524947505774145, 3.88801436995242210653300818263, 3.89509090474010465262003198840, 4.54786764157570828730816389595, 4.58721714779760185124417103057, 4.73284474494995545118946160614, 4.90555867537011437763193333636, 5.57191954976078894604280606031, 5.73015874903168392778394292830, 5.88932420505269698424430521665, 5.90596707384285186231098178335, 6.48774372460547012649762794703, 6.60171800811418718193766658615, 6.76354037176925662676899740126, 6.91043872904680642404314832304, 7.22715189671199119391494709357, 7.51142627490353454858613080239
