| L(s) = 1 | − 4·5-s − 6·9-s − 2·17-s + 5·25-s − 10·29-s + 12·37-s − 8·41-s + 24·45-s − 14·49-s − 14·53-s + 10·61-s − 16·73-s + 27·81-s + 8·85-s + 32·89-s + 16·97-s − 2·101-s + 40·109-s − 14·113-s − 22·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 40·145-s + 149-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2·9-s − 0.485·17-s + 25-s − 1.85·29-s + 1.97·37-s − 1.24·41-s + 3.57·45-s − 2·49-s − 1.92·53-s + 1.28·61-s − 1.87·73-s + 3·81-s + 0.867·85-s + 3.39·89-s + 1.62·97-s − 0.199·101-s + 3.83·109-s − 1.31·113-s − 2·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.32·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29246464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5867179880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5867179880\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.047267363032367249238689036218, −8.009443536192604823779096498288, −7.70276162878071649176394814594, −7.66454290419827783715394538691, −6.76154468922796015742242014365, −6.71140093837909891543444052754, −6.18626943713450779423155043873, −5.84596837556311646541770333038, −5.48831457194904525585714366812, −5.06271233566061882317080150927, −4.49591275935552160989500507141, −4.47767317041287295168331492622, −3.67080431120527926069607542902, −3.57625342560398635037056326806, −3.02643055368253838528120263197, −2.93338382182881838519913474230, −1.95192677058092351707406748068, −1.91554975604067973069524470670, −0.66691342743865919612805953123, −0.30897783444820330366712468730,
0.30897783444820330366712468730, 0.66691342743865919612805953123, 1.91554975604067973069524470670, 1.95192677058092351707406748068, 2.93338382182881838519913474230, 3.02643055368253838528120263197, 3.57625342560398635037056326806, 3.67080431120527926069607542902, 4.47767317041287295168331492622, 4.49591275935552160989500507141, 5.06271233566061882317080150927, 5.48831457194904525585714366812, 5.84596837556311646541770333038, 6.18626943713450779423155043873, 6.71140093837909891543444052754, 6.76154468922796015742242014365, 7.66454290419827783715394538691, 7.70276162878071649176394814594, 8.009443536192604823779096498288, 8.047267363032367249238689036218
