| L(s) = 1 | + 3·5-s − 8·11-s + 5·13-s − 5·17-s − 8·19-s + 23-s + 5·25-s + 3·29-s − 8·31-s + 4·37-s − 5·41-s − 11·43-s + 5·47-s − 14·49-s − 9·53-s − 24·55-s − 13·59-s + 61-s + 15·65-s − 5·67-s − 71-s + 9·73-s + 17·79-s + 32·83-s − 15·85-s + 3·89-s − 24·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 2.41·11-s + 1.38·13-s − 1.21·17-s − 1.83·19-s + 0.208·23-s + 25-s + 0.557·29-s − 1.43·31-s + 0.657·37-s − 0.780·41-s − 1.67·43-s + 0.729·47-s − 2·49-s − 1.23·53-s − 3.23·55-s − 1.69·59-s + 0.128·61-s + 1.86·65-s − 0.610·67-s − 0.118·71-s + 1.05·73-s + 1.91·79-s + 3.51·83-s − 1.62·85-s + 0.317·89-s − 2.46·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040211235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040211235\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 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
−9.003434351026882252085404861099, −8.546191163925563324210522558179, −8.370674379851304223787298597149, −7.85437099243471296079175975376, −7.74318730732949618549262307842, −7.00134717004277172353344143879, −6.42799176659838320168571650820, −6.40253052260283404219307950030, −6.09357784928277278171943630680, −5.56117601873884063399741127627, −4.97480591990136706566858905130, −4.88939014141834088097914918072, −4.54993378919572152849517390832, −3.57740626301042166320578420215, −3.45133329499950360906809124492, −2.75225708006681505704631211792, −2.14300460157535248962369625169, −2.08506452467576889631512769217, −1.43697213796588012672250153321, −0.30614336638079191187775652499,
0.30614336638079191187775652499, 1.43697213796588012672250153321, 2.08506452467576889631512769217, 2.14300460157535248962369625169, 2.75225708006681505704631211792, 3.45133329499950360906809124492, 3.57740626301042166320578420215, 4.54993378919572152849517390832, 4.88939014141834088097914918072, 4.97480591990136706566858905130, 5.56117601873884063399741127627, 6.09357784928277278171943630680, 6.40253052260283404219307950030, 6.42799176659838320168571650820, 7.00134717004277172353344143879, 7.74318730732949618549262307842, 7.85437099243471296079175975376, 8.370674379851304223787298597149, 8.546191163925563324210522558179, 9.003434351026882252085404861099
