L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·5-s + 4·6-s + 2·7-s + 4·8-s + 3·9-s + 8·10-s + 6·12-s + 4·14-s + 8·15-s + 5·16-s + 6·18-s − 4·19-s + 12·20-s + 4·21-s − 2·23-s + 8·24-s + 2·25-s + 4·27-s + 6·28-s − 4·29-s + 16·30-s + 4·31-s + 6·32-s + 8·35-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.78·5-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s + 2.52·10-s + 1.73·12-s + 1.06·14-s + 2.06·15-s + 5/4·16-s + 1.41·18-s − 0.917·19-s + 2.68·20-s + 0.872·21-s − 0.417·23-s + 1.63·24-s + 2/5·25-s + 0.769·27-s + 1.13·28-s − 0.742·29-s + 2.92·30-s + 0.718·31-s + 1.06·32-s + 1.35·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(12.46211970\) |
\(L(\frac12)\) |
\(\approx\) |
\(12.46211970\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 14 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.26124258674917254550487217205, −9.738192136252859605356064269867, −9.584007996699129219946956093419, −8.906248210020816594514505143451, −8.605673790472952672294072178775, −8.139633493422719129118916760252, −7.52688652905998670386864167423, −7.37588311853937904325025269121, −6.53318796392168469508655334504, −6.38385247650647585026092836777, −5.81434742854424470904257199085, −5.49619527580230857308832085786, −4.87957428627708397509522203188, −4.58425400030105124017947836583, −3.82024851165530888117705794237, −3.63173394894822702705589040126, −2.60336449179103990576532404784, −2.49878402826380940678274663916, −1.67695778472700422371539025888, −1.63244256544766525743959276876,
1.63244256544766525743959276876, 1.67695778472700422371539025888, 2.49878402826380940678274663916, 2.60336449179103990576532404784, 3.63173394894822702705589040126, 3.82024851165530888117705794237, 4.58425400030105124017947836583, 4.87957428627708397509522203188, 5.49619527580230857308832085786, 5.81434742854424470904257199085, 6.38385247650647585026092836777, 6.53318796392168469508655334504, 7.37588311853937904325025269121, 7.52688652905998670386864167423, 8.139633493422719129118916760252, 8.605673790472952672294072178775, 8.906248210020816594514505143451, 9.584007996699129219946956093419, 9.738192136252859605356064269867, 10.26124258674917254550487217205