| L(s) = 1 | + 284·25-s − 940·49-s + 1.28e3·73-s − 2.29e3·97-s + 5.26e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 2.27·25-s − 2.74·49-s + 2.06·73-s − 2.40·97-s + 3.95·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-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 + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.197848007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197848007\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 470 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2630 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 1150 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 54682 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 38446 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 103430 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 322 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 890822 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 363274 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 574 T + p^{3} T^{2} )^{4} \) |
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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
−6.66342145460443092164477007017, −6.45792741759169404558737740631, −6.19473252602108113973153109669, −5.96649615572181673186670565505, −5.72692000074153191587926859085, −5.39774657732659590211850382896, −5.31796079151300375996184618699, −4.99262636771099118916621861472, −4.80054530558251278486492659496, −4.48461344444376810290787022169, −4.43200511806724335986078499005, −4.23314817664151869032207861889, −3.61358475823917058532946860367, −3.52243760349827180664154857187, −3.32033069383763134474198859438, −3.08922685911511800726255743696, −2.81056291967250817191022817103, −2.39697541741472857782276049880, −2.28398610328640160690189559074, −1.84192360072070073976371195044, −1.57775885256110712766981083427, −1.16599054570433810125257267172, −0.982462127589477152619674389567, −0.57284093945999394678194049924, −0.18634918642795975289557131741,
0.18634918642795975289557131741, 0.57284093945999394678194049924, 0.982462127589477152619674389567, 1.16599054570433810125257267172, 1.57775885256110712766981083427, 1.84192360072070073976371195044, 2.28398610328640160690189559074, 2.39697541741472857782276049880, 2.81056291967250817191022817103, 3.08922685911511800726255743696, 3.32033069383763134474198859438, 3.52243760349827180664154857187, 3.61358475823917058532946860367, 4.23314817664151869032207861889, 4.43200511806724335986078499005, 4.48461344444376810290787022169, 4.80054530558251278486492659496, 4.99262636771099118916621861472, 5.31796079151300375996184618699, 5.39774657732659590211850382896, 5.72692000074153191587926859085, 5.96649615572181673186670565505, 6.19473252602108113973153109669, 6.45792741759169404558737740631, 6.66342145460443092164477007017
