| L(s) = 1 | + 3.26e3·9-s − 1.19e6·17-s + 3.90e6·25-s + 6.86e7·41-s − 8.07e7·49-s + 8.44e8·73-s − 3.76e8·81-s − 2.17e9·89-s − 3.47e9·97-s + 2.29e9·113-s + 3.54e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.90e9·153-s + 157-s + 163-s + 167-s + 2.12e10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 0.165·9-s − 3.47·17-s + 2·25-s + 3.79·41-s − 2·49-s + 3.48·73-s − 0.972·81-s − 3.68·89-s − 3.98·97-s + 1.32·113-s + 0.150·121-s − 0.575·153-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.524857123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.524857123\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{18} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 354349618 T^{2} + p^{18} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 597510 T + p^{9} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 335013705758 T^{2} + p^{18} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 34306362 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 1000250360894414 T^{2} + p^{18} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10228968070290322 T^{2} + p^{18} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 50467064407716994 T^{2} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 422324930 T + p^{9} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 352960460737558306 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1089849006 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1738254710 T + p^{9} T^{2} )^{2} \) |
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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
−13.03053767191208780186927824832, −12.84807370069876657290302759418, −12.44569336412597739285527916548, −11.17959200870430913064695055609, −11.10182810663012250082376202368, −10.83605369447645095706114592366, −9.646495822014850831218031628157, −9.309392875949209074459744187050, −8.647711668414114287852455210837, −8.185301553968613984526440695682, −7.16589091104376074649291661184, −6.71951468053183707218970546615, −6.22661171593006831491305194470, −5.19643130760178539449584535426, −4.43973078790012342746311078659, −4.11444714593756977299677698945, −2.76251509808669956280400975647, −2.37513433198687195154446743481, −1.33148871414915706942241167858, −0.38656767958788163960738348630,
0.38656767958788163960738348630, 1.33148871414915706942241167858, 2.37513433198687195154446743481, 2.76251509808669956280400975647, 4.11444714593756977299677698945, 4.43973078790012342746311078659, 5.19643130760178539449584535426, 6.22661171593006831491305194470, 6.71951468053183707218970546615, 7.16589091104376074649291661184, 8.185301553968613984526440695682, 8.647711668414114287852455210837, 9.309392875949209074459744187050, 9.646495822014850831218031628157, 10.83605369447645095706114592366, 11.10182810663012250082376202368, 11.17959200870430913064695055609, 12.44569336412597739285527916548, 12.84807370069876657290302759418, 13.03053767191208780186927824832
