| L(s) = 1 | + 2-s − 3·3-s − 3·6-s + 7-s − 8-s + 6·9-s − 3·11-s − 7·13-s + 14-s − 16-s + 3·17-s + 6·18-s + 7·19-s − 3·21-s − 3·22-s + 12·23-s + 3·24-s − 2·25-s − 7·26-s − 9·27-s − 3·29-s + 8·31-s + 9·33-s + 3·34-s − 12·37-s + 7·38-s + 21·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 1.22·6-s + 0.377·7-s − 0.353·8-s + 2·9-s − 0.904·11-s − 1.94·13-s + 0.267·14-s − 1/4·16-s + 0.727·17-s + 1.41·18-s + 1.60·19-s − 0.654·21-s − 0.639·22-s + 2.50·23-s + 0.612·24-s − 2/5·25-s − 1.37·26-s − 1.73·27-s − 0.557·29-s + 1.43·31-s + 1.56·33-s + 0.514·34-s − 1.97·37-s + 1.13·38-s + 3.36·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.064645173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.064645173\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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
−10.89043586202233015666303417298, −10.82196633687109774518518307236, −10.34460179827228571772165423992, −9.739969867583760169079226917191, −9.448129594203932023987951652966, −9.084645422554149207647711879063, −8.122210821598077554277622519997, −7.57271894389110726842678646524, −7.38812972892025911663621270373, −6.94855126301734884961396599558, −6.33047353061101564276899378472, −5.69686243724189067431138911879, −5.28994329213882025068119788345, −5.04973931195115723305315913715, −4.74483735895783892611478179094, −4.15665203548347950283258225297, −2.98005448757688447831486652410, −2.92415895775985090378529506882, −1.58836594307095652753819332374, −0.61989339783572163750365976131,
0.61989339783572163750365976131, 1.58836594307095652753819332374, 2.92415895775985090378529506882, 2.98005448757688447831486652410, 4.15665203548347950283258225297, 4.74483735895783892611478179094, 5.04973931195115723305315913715, 5.28994329213882025068119788345, 5.69686243724189067431138911879, 6.33047353061101564276899378472, 6.94855126301734884961396599558, 7.38812972892025911663621270373, 7.57271894389110726842678646524, 8.122210821598077554277622519997, 9.084645422554149207647711879063, 9.448129594203932023987951652966, 9.739969867583760169079226917191, 10.34460179827228571772165423992, 10.82196633687109774518518307236, 10.89043586202233015666303417298
