L(s) = 1 | − 2·3-s + 4-s + 5-s + 3·9-s + 4·11-s − 2·12-s − 2·15-s + 16-s + 20-s − 2·25-s − 4·27-s − 8·33-s + 3·36-s − 4·37-s + 4·44-s + 3·45-s + 16·47-s − 2·48-s + 2·49-s − 4·53-s + 4·55-s + 8·59-s − 2·60-s + 64-s − 8·67-s + 16·71-s + 4·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.447·5-s + 9-s + 1.20·11-s − 0.577·12-s − 0.516·15-s + 1/4·16-s + 0.223·20-s − 2/5·25-s − 0.769·27-s − 1.39·33-s + 1/2·36-s − 0.657·37-s + 0.603·44-s + 0.447·45-s + 2.33·47-s − 0.288·48-s + 2/7·49-s − 0.549·53-s + 0.539·55-s + 1.04·59-s − 0.258·60-s + 1/8·64-s − 0.977·67-s + 1.89·71-s + 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.049184077\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.049184077\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.83812995438233265341033341820, −10.41137331151424430082490492183, −9.803064351421850226496301995439, −9.329172792456276416019531911476, −8.754931371176731833819107492896, −7.976194554773426566810206855329, −7.21201674050179372835015480714, −6.82378435606469854538163155443, −6.24843804240280296624777703161, −5.72945068915460235250490124046, −5.19439406411647697870324199977, −4.26893653779922028883425571731, −3.67813478878468384844838485841, −2.37891002330605190710787923901, −1.28513317176888455792170575842,
1.28513317176888455792170575842, 2.37891002330605190710787923901, 3.67813478878468384844838485841, 4.26893653779922028883425571731, 5.19439406411647697870324199977, 5.72945068915460235250490124046, 6.24843804240280296624777703161, 6.82378435606469854538163155443, 7.21201674050179372835015480714, 7.976194554773426566810206855329, 8.754931371176731833819107492896, 9.329172792456276416019531911476, 9.803064351421850226496301995439, 10.41137331151424430082490492183, 10.83812995438233265341033341820