| L(s) = 1 | − 8·7-s − 18·9-s − 40·17-s − 144·23-s + 148·25-s + 872·31-s + 8·41-s + 848·47-s + 684·49-s + 144·63-s − 1.13e3·71-s − 728·73-s − 1.24e3·79-s + 243·81-s + 2.74e3·89-s + 3.56e3·97-s + 1.19e3·103-s − 184·113-s + 320·119-s + 4.14e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 720·153-s + ⋯ |
| L(s) = 1 | − 0.431·7-s − 2/3·9-s − 0.570·17-s − 1.30·23-s + 1.18·25-s + 5.05·31-s + 0.0304·41-s + 2.63·47-s + 1.99·49-s + 0.287·63-s − 1.89·71-s − 1.16·73-s − 1.76·79-s + 1/3·81-s + 3.26·89-s + 3.72·97-s + 1.14·103-s − 0.153·113-s + 0.246·119-s + 3.11·121-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 + 0.380·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.066957180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.066957180\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 8054 T^{4} - 148 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 4 T - 318 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4140 T^{2} + 7569974 T^{4} - 4140 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2316 T^{2} + 643990 T^{4} - 2316 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 20 T + 9478 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4524 T^{2} + 243046 p^{2} T^{4} - 4524 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 72 T + 25182 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4844 T^{2} + 653432726 T^{4} + 4844 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 436 T + 104306 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 176876 T^{2} + 12810225654 T^{4} - 176876 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T - 23882 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 3300 p T^{2} + 17519620726 T^{4} - 3300 p^{7} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 424 T + 176878 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 171572 T^{2} + 23330484726 T^{4} - 171572 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 665580 T^{2} + 191637957494 T^{4} - 665580 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 698252 T^{2} + 220734819798 T^{4} - 698252 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 704588 T^{2} + 298087367766 T^{4} - 704588 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 p T + 796030 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 364 T + 666006 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 620 T + 665426 T^{2} + 620 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 875660 T^{2} + 779244875286 T^{4} - 875660 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1372 T + 1477334 T^{2} - 1372 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 1780 T + 2572646 T^{2} - 1780 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 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
−7.07667612351902183595254910825, −6.52286692361827567297146931659, −6.40856127501689536652271392579, −6.38519145366012455828080933288, −6.33739436928912185222082768098, −5.68939750559778079692321979862, −5.60835488029743505694982377255, −5.56337973411253679708827733670, −5.17643336341342089838958895171, −4.61502192250844152862428901411, −4.54456400292843395673259686935, −4.31999758788939255041318108967, −4.20143675063246084820505463321, −4.07846010486851575567480115114, −3.19874086544243157357944473669, −3.19686625132701952660132653442, −2.95135453607277317818511736165, −2.90727691406168372655580844211, −2.26248345263371857864896489665, −2.05538340860121917975491257556, −1.98560366484110223661174795622, −1.20531250264403527157759482279, −0.68270313095361190785457282839, −0.68257542838983771342863953664, −0.54384550937677722332364257672,
0.54384550937677722332364257672, 0.68257542838983771342863953664, 0.68270313095361190785457282839, 1.20531250264403527157759482279, 1.98560366484110223661174795622, 2.05538340860121917975491257556, 2.26248345263371857864896489665, 2.90727691406168372655580844211, 2.95135453607277317818511736165, 3.19686625132701952660132653442, 3.19874086544243157357944473669, 4.07846010486851575567480115114, 4.20143675063246084820505463321, 4.31999758788939255041318108967, 4.54456400292843395673259686935, 4.61502192250844152862428901411, 5.17643336341342089838958895171, 5.56337973411253679708827733670, 5.60835488029743505694982377255, 5.68939750559778079692321979862, 6.33739436928912185222082768098, 6.38519145366012455828080933288, 6.40856127501689536652271392579, 6.52286692361827567297146931659, 7.07667612351902183595254910825
