| L(s) = 1 | − 2·3-s − 4-s + 5-s + 7-s + 3·9-s + 11-s + 2·12-s − 13-s − 2·15-s + 2·17-s − 20-s − 2·21-s − 4·27-s − 28-s − 2·29-s − 2·33-s + 35-s − 3·36-s + 2·39-s − 44-s + 3·45-s − 47-s − 4·51-s + 52-s + 55-s + 2·60-s + 3·63-s + ⋯ |
| L(s) = 1 | − 2·3-s − 4-s + 5-s + 7-s + 3·9-s + 11-s + 2·12-s − 13-s − 2·15-s + 2·17-s − 20-s − 2·21-s − 4·27-s − 28-s − 2·29-s − 2·33-s + 35-s − 3·36-s + 2·39-s − 44-s + 3·45-s − 47-s − 4·51-s + 52-s + 55-s + 2·60-s + 3·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3752517378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3752517378\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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
−12.03701092219879091315253732064, −11.52990567531916683707658013002, −11.42705320179071613570104099214, −10.71771737634813806185965207332, −10.25632170110440673772710657933, −9.809499518364558921841200811466, −9.379129554727138987262181592000, −9.330075399939161905805593651989, −8.230843306843809813624834981892, −7.74128898570715815818246340046, −7.21199101013761836299966820716, −6.73398311178070495164713305563, −5.88403537497648354489194189545, −5.76417212870445232069862411033, −5.05318271691171487941791541812, −4.93680935064037131939321005328, −4.17047827773770065875382802897, −3.60097272772237206383982674068, −1.94850255007359042452430354403, −1.30636796685490247482543477442,
1.30636796685490247482543477442, 1.94850255007359042452430354403, 3.60097272772237206383982674068, 4.17047827773770065875382802897, 4.93680935064037131939321005328, 5.05318271691171487941791541812, 5.76417212870445232069862411033, 5.88403537497648354489194189545, 6.73398311178070495164713305563, 7.21199101013761836299966820716, 7.74128898570715815818246340046, 8.230843306843809813624834981892, 9.330075399939161905805593651989, 9.379129554727138987262181592000, 9.809499518364558921841200811466, 10.25632170110440673772710657933, 10.71771737634813806185965207332, 11.42705320179071613570104099214, 11.52990567531916683707658013002, 12.03701092219879091315253732064
