| L(s) = 1 | + 4·3-s − 6·4-s − 42·9-s − 24·12-s − 101·16-s − 332·25-s − 292·27-s + 328·31-s + 252·36-s + 752·37-s − 404·48-s + 1.17e3·49-s + 1.04e3·64-s + 2.40e3·67-s − 1.32e3·75-s + 971·81-s + 1.31e3·93-s + 952·97-s + 1.99e3·100-s − 7.07e3·103-s + 1.75e3·108-s + 3.00e3·111-s − 1.96e3·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 3/4·4-s − 1.55·9-s − 0.577·12-s − 1.57·16-s − 2.65·25-s − 2.08·27-s + 1.90·31-s + 7/6·36-s + 3.34·37-s − 1.21·48-s + 24/7·49-s + 2.03·64-s + 4.39·67-s − 2.04·75-s + 1.33·81-s + 1.46·93-s + 0.996·97-s + 1.99·100-s − 6.76·103-s + 1.56·108-s + 2.57·111-s − 1.42·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.965978912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.965978912\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 166 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 12 p^{2} T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3944 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 1874 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 10676 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 15910 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 14922 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 188 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 137374 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 68276 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 137342 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 267698 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 329222 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 419640 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 602 T + p^{3} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 234926 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 685584 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 952796 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 943686 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 567538 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 238 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.980509635409259325294204846991, −7.76159321512454467419373307243, −7.52696870923059752775596848709, −6.94757333078756227324056768102, −6.89622524679939913596647103997, −6.50320015162175507091867688028, −6.44295680306766210043266333221, −5.79037567062639961726055629863, −5.75620985950249744611824930997, −5.66596154293320759216589504651, −5.29892353167986715135909392784, −4.93055739164492635584057407508, −4.53010407305067828134651669933, −4.21296653191924957579263949506, −4.05853062494587610018361806666, −3.89473338438033855865326201805, −3.55054780889257454620559196720, −2.90646122268931725973433186546, −2.66542440725793330748734126777, −2.46623384157991914161028486502, −2.25751601370671067481022300715, −1.87252039660796404736006548699, −1.04270828095698483227463681982, −0.48294317335395388418325947615, −0.47806115228620977832455141551,
0.47806115228620977832455141551, 0.48294317335395388418325947615, 1.04270828095698483227463681982, 1.87252039660796404736006548699, 2.25751601370671067481022300715, 2.46623384157991914161028486502, 2.66542440725793330748734126777, 2.90646122268931725973433186546, 3.55054780889257454620559196720, 3.89473338438033855865326201805, 4.05853062494587610018361806666, 4.21296653191924957579263949506, 4.53010407305067828134651669933, 4.93055739164492635584057407508, 5.29892353167986715135909392784, 5.66596154293320759216589504651, 5.75620985950249744611824930997, 5.79037567062639961726055629863, 6.44295680306766210043266333221, 6.50320015162175507091867688028, 6.89622524679939913596647103997, 6.94757333078756227324056768102, 7.52696870923059752775596848709, 7.76159321512454467419373307243, 7.980509635409259325294204846991
