| L(s) = 1 | − 2·4-s − 2·25-s + 16·31-s + 16·49-s + 8·64-s + 48·71-s − 16·79-s − 24·89-s + 4·100-s + 20·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 32·196-s + ⋯ |
| L(s) = 1 | − 4-s − 2/5·25-s + 2.87·31-s + 16/7·49-s + 64-s + 5.69·71-s − 1.80·79-s − 2.54·89-s + 2/5·100-s + 1.81·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-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 − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 2.28·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.434530147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.434530147\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.357016370286650825145375712662, −8.140052446631286621262658391081, −8.007220174594478590002297617751, −7.48534017911455403144071472684, −7.23313098053844548275720731489, −7.20312082287549614842798250597, −6.70141118107415130971823404753, −6.64641319604065193482331579144, −6.03245287575179607252013064012, −5.99660398049163292815948163299, −5.97418809060218922823645102518, −5.24004343303546215608186826086, −5.09215900525180567985896061918, −4.84261841307438770111456237884, −4.77462818805727155274658996389, −4.16757716991330336580032420465, −4.00230471430741844087964481399, −3.65335937704852227453666852293, −3.64399532427296982381799044059, −2.73956007242118671525097618182, −2.54240462806383525162049075251, −2.51617530039203534807036253658, −1.71115904636216459061935148406, −1.09477466940388879861630604778, −0.59181828826030600195408820638,
0.59181828826030600195408820638, 1.09477466940388879861630604778, 1.71115904636216459061935148406, 2.51617530039203534807036253658, 2.54240462806383525162049075251, 2.73956007242118671525097618182, 3.64399532427296982381799044059, 3.65335937704852227453666852293, 4.00230471430741844087964481399, 4.16757716991330336580032420465, 4.77462818805727155274658996389, 4.84261841307438770111456237884, 5.09215900525180567985896061918, 5.24004343303546215608186826086, 5.97418809060218922823645102518, 5.99660398049163292815948163299, 6.03245287575179607252013064012, 6.64641319604065193482331579144, 6.70141118107415130971823404753, 7.20312082287549614842798250597, 7.23313098053844548275720731489, 7.48534017911455403144071472684, 8.007220174594478590002297617751, 8.140052446631286621262658391081, 8.357016370286650825145375712662
