| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s + 12-s + 4·13-s + 15-s − 16-s + 4·17-s − 18-s + 8·19-s + 20-s − 3·24-s + 25-s − 4·26-s − 27-s + 4·29-s − 30-s − 5·32-s − 4·34-s − 36-s + 20·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.10·13-s + 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.612·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.742·29-s − 0.182·30-s − 0.883·32-s − 0.685·34-s − 1/6·36-s + 3.28·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9303941189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9303941189\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−9.327820441539330420804608673051, −8.455181921164028207665051683361, −8.074746559817531846840486780032, −7.67749235610857150327119620769, −7.46261028773657489832215471574, −6.56365355473852823699216167747, −6.20681526904542088835730053039, −5.63406482120703248001185629214, −4.87700802399738744801917392626, −4.79492945983632365647604814265, −3.76558570458913747879946495025, −3.55714635244582915245852508954, −2.61544872003966305633910980827, −1.27025970316177507740293768624, −0.870154925974925884967934945032,
0.870154925974925884967934945032, 1.27025970316177507740293768624, 2.61544872003966305633910980827, 3.55714635244582915245852508954, 3.76558570458913747879946495025, 4.79492945983632365647604814265, 4.87700802399738744801917392626, 5.63406482120703248001185629214, 6.20681526904542088835730053039, 6.56365355473852823699216167747, 7.46261028773657489832215471574, 7.67749235610857150327119620769, 8.074746559817531846840486780032, 8.455181921164028207665051683361, 9.327820441539330420804608673051
