| L(s) = 1 | − 4-s + 16-s + 8·19-s − 12·29-s + 16·31-s + 12·41-s − 2·49-s − 20·61-s − 64-s − 8·76-s − 16·79-s + 36·89-s − 36·101-s + 20·109-s + 12·116-s − 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 12·164-s + 167-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s + 1.83·19-s − 2.22·29-s + 2.87·31-s + 1.87·41-s − 2/7·49-s − 2.56·61-s − 1/8·64-s − 0.917·76-s − 1.80·79-s + 3.81·89-s − 3.58·101-s + 1.91·109-s + 1.11·116-s − 2·121-s − 1.43·124-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 − 0.937·164-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.498074284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498074284\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−11.30299090140795773444738240252, −10.92364972653550439038892589587, −10.29924749053468720653084671664, −9.981668100371647146233798173812, −9.310092617995571481528373302408, −9.305081061958847708933319532503, −8.734518401528348357812219131052, −7.921454892623089513938599914123, −7.72399626762213868515367902971, −7.42310713674311558565678035377, −6.53514944981278129352164932106, −6.21704172593643872004407751652, −5.45315140469871906146730803665, −5.27976105288632726974539227014, −4.34799478233135075827123058536, −4.19385360768860040961610482192, −3.14981777812299502186399256676, −2.90916387940588836638728492533, −1.76826063882083140168211212390, −0.821288108852431930051466738368,
0.821288108852431930051466738368, 1.76826063882083140168211212390, 2.90916387940588836638728492533, 3.14981777812299502186399256676, 4.19385360768860040961610482192, 4.34799478233135075827123058536, 5.27976105288632726974539227014, 5.45315140469871906146730803665, 6.21704172593643872004407751652, 6.53514944981278129352164932106, 7.42310713674311558565678035377, 7.72399626762213868515367902971, 7.921454892623089513938599914123, 8.734518401528348357812219131052, 9.305081061958847708933319532503, 9.310092617995571481528373302408, 9.981668100371647146233798173812, 10.29924749053468720653084671664, 10.92364972653550439038892589587, 11.30299090140795773444738240252
