| L(s) = 1 | − 4·7-s + 2·19-s + 8·25-s − 20·29-s + 20·41-s − 24·43-s − 2·49-s + 20·53-s + 24·59-s + 16·61-s + 16·71-s + 12·73-s + 12·89-s + 16·107-s + 12·113-s + 4·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.458·19-s + 8/5·25-s − 3.71·29-s + 3.12·41-s − 3.65·43-s − 2/7·49-s + 2.74·53-s + 3.12·59-s + 2.04·61-s + 1.89·71-s + 1.40·73-s + 1.27·89-s + 1.54·107-s + 1.12·113-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-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 + 1.38·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.640104843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.640104843\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 148 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−8.906418544850366507911922198744, −8.704162875101944710790112934832, −8.453744924674234598200545653154, −7.71865709717719150320283949744, −7.47618608457496133074704355911, −7.04931526000035031761977189596, −6.79080499329829796541042616820, −6.40831483395080228282414480273, −5.97859252613812734879612858794, −5.49906396392519123990776075042, −5.14355431780296368447505403770, −4.96257444134477495286624829039, −3.89347823598642255882414816154, −3.78551198529710502008390937481, −3.56230926850487348522793878462, −2.95036380937654436880837791093, −2.19305119089552562109018607698, −2.16636562265307172176770250809, −1.02033610835468270570295208608, −0.48466762490716200556038009173,
0.48466762490716200556038009173, 1.02033610835468270570295208608, 2.16636562265307172176770250809, 2.19305119089552562109018607698, 2.95036380937654436880837791093, 3.56230926850487348522793878462, 3.78551198529710502008390937481, 3.89347823598642255882414816154, 4.96257444134477495286624829039, 5.14355431780296368447505403770, 5.49906396392519123990776075042, 5.97859252613812734879612858794, 6.40831483395080228282414480273, 6.79080499329829796541042616820, 7.04931526000035031761977189596, 7.47618608457496133074704355911, 7.71865709717719150320283949744, 8.453744924674234598200545653154, 8.704162875101944710790112934832, 8.906418544850366507911922198744
