| L(s) = 1 | + 3·3-s − 5-s + 3·9-s + 5·11-s − 10·13-s − 3·15-s + 7·17-s + 2·19-s + 2·23-s + 14·29-s − 4·31-s + 15·33-s + 6·37-s − 30·39-s − 24·41-s − 4·43-s − 3·45-s − 47-s + 21·51-s − 5·55-s + 6·57-s + 4·59-s − 4·61-s + 10·65-s − 8·67-s + 6·69-s − 6·73-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 9-s + 1.50·11-s − 2.77·13-s − 0.774·15-s + 1.69·17-s + 0.458·19-s + 0.417·23-s + 2.59·29-s − 0.718·31-s + 2.61·33-s + 0.986·37-s − 4.80·39-s − 3.74·41-s − 0.609·43-s − 0.447·45-s − 0.145·47-s + 2.94·51-s − 0.674·55-s + 0.794·57-s + 0.520·59-s − 0.512·61-s + 1.24·65-s − 0.977·67-s + 0.722·69-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.842995238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.842995238\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 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
−9.592585685824871422610944124306, −8.991529132564081315530590393352, −8.527570323488784587224997266204, −8.111701066728360204540925229333, −8.085852170055826710970111404176, −7.40723732940083477675952570228, −7.19847931407547598390604732534, −6.80217757212807696289372078710, −6.45425698652476177484518995146, −5.72806818462458904720266097681, −5.17349007321820450794720363323, −4.81309465552437004620917539257, −4.51523193485795164904140398988, −3.84157475150129367150184251141, −3.31542343025959803408636589043, −2.95401181597031424857226848762, −2.87856913507056027153866634125, −1.99448206124489268171465159183, −1.55336165259354434464637005554, −0.63248053491435312529726403511,
0.63248053491435312529726403511, 1.55336165259354434464637005554, 1.99448206124489268171465159183, 2.87856913507056027153866634125, 2.95401181597031424857226848762, 3.31542343025959803408636589043, 3.84157475150129367150184251141, 4.51523193485795164904140398988, 4.81309465552437004620917539257, 5.17349007321820450794720363323, 5.72806818462458904720266097681, 6.45425698652476177484518995146, 6.80217757212807696289372078710, 7.19847931407547598390604732534, 7.40723732940083477675952570228, 8.085852170055826710970111404176, 8.111701066728360204540925229333, 8.527570323488784587224997266204, 8.991529132564081315530590393352, 9.592585685824871422610944124306
