| L(s) = 1 | − 2·2-s + 3·4-s − 3·5-s − 4·7-s − 4·8-s − 3·9-s + 6·10-s + 3·11-s + 13-s + 8·14-s + 5·16-s − 3·17-s + 6·18-s + 7·19-s − 9·20-s − 6·22-s + 9·23-s + 5·25-s − 2·26-s − 12·28-s − 3·29-s + 16·31-s − 6·32-s + 6·34-s + 12·35-s − 9·36-s + 37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.34·5-s − 1.51·7-s − 1.41·8-s − 9-s + 1.89·10-s + 0.904·11-s + 0.277·13-s + 2.13·14-s + 5/4·16-s − 0.727·17-s + 1.41·18-s + 1.60·19-s − 2.01·20-s − 1.27·22-s + 1.87·23-s + 25-s − 0.392·26-s − 2.26·28-s − 0.557·29-s + 2.87·31-s − 1.06·32-s + 1.02·34-s + 2.02·35-s − 3/2·36-s + 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3535492229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3535492229\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.15432927235022881694464068787, −12.97546810616720897735671154607, −12.40394875032453460772703553711, −11.78867811891303595147579852749, −11.44217736124206367817293983731, −11.28783340494978169916094368559, −10.41558855417257958818082274299, −9.871231791840324108513352673647, −9.364205306908958065473550705383, −8.807739756753312397387325094851, −8.563544964422620653554016335354, −7.77319976749745855321281860600, −7.24072744906191133014276375488, −6.61474407336360658553355825354, −6.31989551707620819368844786179, −5.31286683815536039308197177838, −4.19774518413936593886313699907, −3.08455353736824444039938934560, −3.02756476173093793568102837816, −0.813715947122070317671206201682,
0.813715947122070317671206201682, 3.02756476173093793568102837816, 3.08455353736824444039938934560, 4.19774518413936593886313699907, 5.31286683815536039308197177838, 6.31989551707620819368844786179, 6.61474407336360658553355825354, 7.24072744906191133014276375488, 7.77319976749745855321281860600, 8.563544964422620653554016335354, 8.807739756753312397387325094851, 9.364205306908958065473550705383, 9.871231791840324108513352673647, 10.41558855417257958818082274299, 11.28783340494978169916094368559, 11.44217736124206367817293983731, 11.78867811891303595147579852749, 12.40394875032453460772703553711, 12.97546810616720897735671154607, 14.15432927235022881694464068787
