| L(s) = 1 | + 100·9-s + 456·17-s + 436·25-s − 1.22e3·49-s + 6.04e3·81-s + 7.27e3·113-s − 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4.56e4·153-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 3.70·9-s + 6.50·17-s + 3.48·25-s − 3.58·49-s + 8.28·81-s + 6.05·113-s − 4·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 24.0·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(24.95051920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(24.95051920\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 218 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 614 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 27830 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 13894 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 111490 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 128554 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 317990 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.14646700667869170074277343000, −6.77035291564498937452429560165, −6.35664858476086211217166763430, −6.29632250044155556394084103502, −6.24335216661523665124391168182, −5.60739322719269310068746873419, −5.34495591419483886972721416974, −5.21904066203221727610461113217, −5.20263459411241994324494870842, −4.72614439732522367276454308747, −4.58447641022976690460712429427, −4.41179300902422430406136931806, −3.96045027542235399739548356545, −3.68115063138233718562954566666, −3.40514440466808035128478991125, −3.24968231482626736720406780360, −3.03470568824294042616519003515, −2.97981169358095153220363573530, −2.15539189522419466948131426337, −1.81199409212303256734610448942, −1.45257892079413220735776525317, −1.20179157497760076143325757524, −1.13411660648558669153442893813, −0.905278196903914937964373120889, −0.55864329336077472517644823545,
0.55864329336077472517644823545, 0.905278196903914937964373120889, 1.13411660648558669153442893813, 1.20179157497760076143325757524, 1.45257892079413220735776525317, 1.81199409212303256734610448942, 2.15539189522419466948131426337, 2.97981169358095153220363573530, 3.03470568824294042616519003515, 3.24968231482626736720406780360, 3.40514440466808035128478991125, 3.68115063138233718562954566666, 3.96045027542235399739548356545, 4.41179300902422430406136931806, 4.58447641022976690460712429427, 4.72614439732522367276454308747, 5.20263459411241994324494870842, 5.21904066203221727610461113217, 5.34495591419483886972721416974, 5.60739322719269310068746873419, 6.24335216661523665124391168182, 6.29632250044155556394084103502, 6.35664858476086211217166763430, 6.77035291564498937452429560165, 7.14646700667869170074277343000
