| L(s) = 1 | − 10·7-s − 38·19-s − 31·25-s + 170·43-s − 23·49-s − 206·61-s − 50·73-s + 233·121-s + 127-s + 131-s + 380·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 338·169-s + 173-s + 310·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.42·7-s − 2·19-s − 1.23·25-s + 3.95·43-s − 0.469·49-s − 3.37·61-s − 0.684·73-s + 1.92·121-s + 0.00787·127-s + 0.00763·131-s + 20/7·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2·169-s + 0.00578·173-s + 1.77·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2339152039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2339152039\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 31 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 233 T^{2} + p^{4} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 353 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 158 T^{2} + p^{4} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 85 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 1207 T^{2} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 103 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 5678 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−9.003375776105538379975515470044, −8.520251113067188930604253029218, −7.88304236515728376305343684040, −7.85835925117282717397027390615, −7.28761087295284872295880595618, −6.95311150665006849501245783642, −6.32554190692933099955530568788, −6.27199671639870443681359395795, −5.82530888837939702203347701769, −5.65495222452374616012664746396, −4.72919061363806585950694896786, −4.49559030205580956321159552821, −4.03186941146479049520641515967, −3.73982750390992094706726659173, −3.02832080621660927754447811540, −2.81179179516235639742363648654, −2.17261856610763729126113087565, −1.76028119264910497727000919794, −0.916247952570792087784906262555, −0.12913435781348971718621030281,
0.12913435781348971718621030281, 0.916247952570792087784906262555, 1.76028119264910497727000919794, 2.17261856610763729126113087565, 2.81179179516235639742363648654, 3.02832080621660927754447811540, 3.73982750390992094706726659173, 4.03186941146479049520641515967, 4.49559030205580956321159552821, 4.72919061363806585950694896786, 5.65495222452374616012664746396, 5.82530888837939702203347701769, 6.27199671639870443681359395795, 6.32554190692933099955530568788, 6.95311150665006849501245783642, 7.28761087295284872295880595618, 7.85835925117282717397027390615, 7.88304236515728376305343684040, 8.520251113067188930604253029218, 9.003375776105538379975515470044
