| L(s) = 1 | − 3·9-s − 4·13-s + 10·25-s − 20·37-s + 2·49-s + 28·61-s + 20·73-s + 9·81-s − 28·97-s − 4·109-s + 12·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s − 1.10·13-s + 2·25-s − 3.28·37-s + 2/7·49-s + 3.58·61-s + 2.34·73-s + 81-s − 2.84·97-s − 0.383·109-s + 1.10·117-s − 2·121-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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6385144647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6385144647\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−15.92369172058273067985032747780, −15.44375138708661754247537486481, −14.60504122510036768778392573017, −14.46956971764483229019315726323, −13.83864131223898957951710912710, −13.12925026034704855482017874281, −12.24056695896816489745402282108, −12.23659147913522161754220310176, −11.26412486568634322020379293105, −10.73331120502494611207295480445, −10.10407073322686381731412723868, −9.338892356501978764549936387737, −8.625402204712350869792689549676, −8.201099013122993350293745119564, −7.02451456717377461766522669630, −6.74418422965018797428766625394, −5.37448383935061293331068662521, −5.06935218801281721891351547462, −3.63537616080026223751227237238, −2.52494733590377164458054694339,
2.52494733590377164458054694339, 3.63537616080026223751227237238, 5.06935218801281721891351547462, 5.37448383935061293331068662521, 6.74418422965018797428766625394, 7.02451456717377461766522669630, 8.201099013122993350293745119564, 8.625402204712350869792689549676, 9.338892356501978764549936387737, 10.10407073322686381731412723868, 10.73331120502494611207295480445, 11.26412486568634322020379293105, 12.23659147913522161754220310176, 12.24056695896816489745402282108, 13.12925026034704855482017874281, 13.83864131223898957951710912710, 14.46956971764483229019315726323, 14.60504122510036768778392573017, 15.44375138708661754247537486481, 15.92369172058273067985032747780
