| L(s) = 1 | + 2·2-s + 2·3-s − 4-s + 2·5-s + 4·6-s + 4·7-s − 8·8-s + 3·9-s + 4·10-s − 8·11-s − 2·12-s + 8·13-s + 8·14-s + 4·15-s − 7·16-s + 6·18-s − 4·19-s − 2·20-s + 8·21-s − 16·22-s + 8·23-s − 16·24-s + 3·25-s + 16·26-s + 4·27-s − 4·28-s + 8·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s − 1/2·4-s + 0.894·5-s + 1.63·6-s + 1.51·7-s − 2.82·8-s + 9-s + 1.26·10-s − 2.41·11-s − 0.577·12-s + 2.21·13-s + 2.13·14-s + 1.03·15-s − 7/4·16-s + 1.41·18-s − 0.917·19-s − 0.447·20-s + 1.74·21-s − 3.41·22-s + 1.66·23-s − 3.26·24-s + 3/5·25-s + 3.13·26-s + 0.769·27-s − 0.755·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36180225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36180225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.30671921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.30671921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 401 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 192 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 32 T + 448 T^{2} - 32 p T^{3} + p^{2} T^{4} \) |
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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
−8.169695054692912607551537364326, −8.084434442654227120434756815495, −7.67894241326368992001723520895, −7.41986403940584697780105272197, −6.48663388732873319491872932914, −6.31665466246614846295740389197, −6.05582846946838743704357532501, −5.64410347443394617679032525300, −5.09739035044118065689555487808, −4.94970394782757100849598583499, −4.59281163531757707138818800330, −4.57557248101395109818385263529, −3.68250924307391899271165733990, −3.61542282608098483556676962870, −2.93988152811205336492728684612, −2.88576957355743515938550551355, −2.22307023067950763395749624781, −1.93662292882704021352176023904, −1.02977341255322462151799157154, −0.73323336178880199095390818613,
0.73323336178880199095390818613, 1.02977341255322462151799157154, 1.93662292882704021352176023904, 2.22307023067950763395749624781, 2.88576957355743515938550551355, 2.93988152811205336492728684612, 3.61542282608098483556676962870, 3.68250924307391899271165733990, 4.57557248101395109818385263529, 4.59281163531757707138818800330, 4.94970394782757100849598583499, 5.09739035044118065689555487808, 5.64410347443394617679032525300, 6.05582846946838743704357532501, 6.31665466246614846295740389197, 6.48663388732873319491872932914, 7.41986403940584697780105272197, 7.67894241326368992001723520895, 8.084434442654227120434756815495, 8.169695054692912607551537364326
