| L(s) = 1 | + 18·5-s − 7·9-s + 32·13-s − 14·17-s + 193·25-s − 64·29-s − 2·37-s − 80·41-s − 126·45-s + 14·53-s + 158·61-s + 576·65-s + 286·73-s − 32·81-s − 252·85-s − 194·89-s − 176·97-s − 30·101-s + 226·109-s − 224·117-s + 233·121-s + 1.56e3·125-s + 127-s + 131-s + 137-s + 139-s − 1.15e3·145-s + ⋯ |
| L(s) = 1 | + 18/5·5-s − 7/9·9-s + 2.46·13-s − 0.823·17-s + 7.71·25-s − 2.20·29-s − 0.0540·37-s − 1.95·41-s − 2.79·45-s + 0.264·53-s + 2.59·61-s + 8.86·65-s + 3.91·73-s − 0.395·81-s − 2.96·85-s − 2.17·89-s − 1.81·97-s − 0.297·101-s + 2.07·109-s − 1.91·117-s + 1.92·121-s + 12.5·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.94·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.655712031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.655712031\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 7 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 9 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 233 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 601 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 697 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1801 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2098 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2807 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4153 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 79 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8857 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 7778 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11257 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13714 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 88 T + p^{2} T^{2} )^{2} \) |
| 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
−9.586977667026604313988344631153, −9.116116238778498487166477031995, −8.702063773669085795394885655405, −8.384310049714808237554195919605, −8.296238855552437280075706780499, −7.06347801162689535995625635487, −6.88219835375703289610344491229, −6.49904914379316730823489490103, −5.96315856884367679319921801699, −5.90885972257986475775909418310, −5.43496385337386292322839302535, −5.27495801789066198970121805617, −4.63721334914154721461737840511, −3.64811339888607467423058071506, −3.54053432155495119843057554194, −2.67761028859706087989820194116, −2.22724778417055994938581691713, −1.80768626290080804621631050135, −1.47787924745714228287174659120, −0.73542588193810707843183435653,
0.73542588193810707843183435653, 1.47787924745714228287174659120, 1.80768626290080804621631050135, 2.22724778417055994938581691713, 2.67761028859706087989820194116, 3.54053432155495119843057554194, 3.64811339888607467423058071506, 4.63721334914154721461737840511, 5.27495801789066198970121805617, 5.43496385337386292322839302535, 5.90885972257986475775909418310, 5.96315856884367679319921801699, 6.49904914379316730823489490103, 6.88219835375703289610344491229, 7.06347801162689535995625635487, 8.296238855552437280075706780499, 8.384310049714808237554195919605, 8.702063773669085795394885655405, 9.116116238778498487166477031995, 9.586977667026604313988344631153
