| L(s) = 1 | − 3-s + 5-s + 3·9-s − 8·11-s − 13-s − 15-s − 3·17-s − 5·23-s + 5·25-s − 8·27-s + 7·29-s − 8·31-s + 8·33-s − 20·37-s + 39-s − 5·41-s + 5·43-s + 3·45-s + 7·47-s − 14·49-s + 3·51-s + 11·53-s − 8·55-s + 3·59-s − 11·61-s − 65-s − 3·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 9-s − 2.41·11-s − 0.277·13-s − 0.258·15-s − 0.727·17-s − 1.04·23-s + 25-s − 1.53·27-s + 1.29·29-s − 1.43·31-s + 1.39·33-s − 3.28·37-s + 0.160·39-s − 0.780·41-s + 0.762·43-s + 0.447·45-s + 1.02·47-s − 2·49-s + 0.420·51-s + 1.51·53-s − 1.07·55-s + 0.390·59-s − 1.40·61-s − 0.124·65-s − 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3679237595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3679237595\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 11 T + 50 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
| 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
−10.04800368011995667787338508486, −9.237646558487937903280207354320, −9.079979190177828778473833537055, −8.307669945689508402974777187549, −8.293518292157244484417720111678, −7.54934212166109867911069315000, −7.37191369638364681275538219714, −6.84513795284359119160243168822, −6.59013187668352070275961208234, −5.93590535090412341829876450541, −5.41327700177055763380948298756, −5.14536921381970786079394773795, −5.04750973980332240690537746064, −4.07850164293246499078536635191, −4.02055316445413955237113444861, −2.86884371824776007030639288782, −2.84189849324166981470907796185, −1.85124304508428952985937574753, −1.66622703928096714027856891014, −0.23789180798198819370455248812,
0.23789180798198819370455248812, 1.66622703928096714027856891014, 1.85124304508428952985937574753, 2.84189849324166981470907796185, 2.86884371824776007030639288782, 4.02055316445413955237113444861, 4.07850164293246499078536635191, 5.04750973980332240690537746064, 5.14536921381970786079394773795, 5.41327700177055763380948298756, 5.93590535090412341829876450541, 6.59013187668352070275961208234, 6.84513795284359119160243168822, 7.37191369638364681275538219714, 7.54934212166109867911069315000, 8.293518292157244484417720111678, 8.307669945689508402974777187549, 9.079979190177828778473833537055, 9.237646558487937903280207354320, 10.04800368011995667787338508486
