| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s − 12-s − 15-s − 16-s − 18-s − 6·19-s + 20-s + 12·23-s + 3·24-s + 25-s + 27-s + 10·29-s + 30-s − 5·32-s − 36-s + 6·38-s − 3·40-s − 10·43-s − 45-s − 12·46-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 − 0.258·15-s − 1/4·16-s − 0.235·18-s − 1.37·19-s + 0.223·20-s + 2.50·23-s + 0.612·24-s + 1/5·25-s + 0.192·27-s + 1.85·29-s + 0.182·30-s − 0.883·32-s − 1/6·36-s + 0.973·38-s − 0.474·40-s − 1.52·43-s − 0.149·45-s − 1.76·46-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
| \(L(1)\) |
\(\approx\) |
\(1.096880035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.096880035\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.892399351428167215131965300837, −8.544030733817935620568279576532, −8.265344842528523210883241623812, −7.902520218753514335027686044494, −7.15209964462953520184720736322, −6.72521606672367475647233337637, −6.53468878473785351919474017241, −5.38535674461546969487184019211, −4.97535280951748912982547092092, −4.50946846798418509258280508548, −3.95532995302561227890190721683, −3.23904594666351099422428199201, −2.66889019243147502869235143114, −1.67986934782382229520815967235, −0.75507799196260966330798213842,
0.75507799196260966330798213842, 1.67986934782382229520815967235, 2.66889019243147502869235143114, 3.23904594666351099422428199201, 3.95532995302561227890190721683, 4.50946846798418509258280508548, 4.97535280951748912982547092092, 5.38535674461546969487184019211, 6.53468878473785351919474017241, 6.72521606672367475647233337637, 7.15209964462953520184720736322, 7.902520218753514335027686044494, 8.265344842528523210883241623812, 8.544030733817935620568279576532, 8.892399351428167215131965300837
