| L(s) = 1 | + 6·3-s + 27·9-s − 36·17-s − 144·23-s − 74·25-s + 108·27-s − 468·29-s − 904·43-s + 622·49-s − 216·51-s + 828·53-s + 844·61-s − 864·69-s − 444·75-s + 1.02e3·79-s + 405·81-s − 2.80e3·87-s − 1.11e3·101-s − 16·103-s + 3.52e3·107-s − 2.26e3·113-s + 1.36e3·121-s + 127-s − 5.42e3·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 0.513·17-s − 1.30·23-s − 0.591·25-s + 0.769·27-s − 2.99·29-s − 3.20·43-s + 1.81·49-s − 0.593·51-s + 2.14·53-s + 1.77·61-s − 1.50·69-s − 0.683·75-s + 1.45·79-s + 5/9·81-s − 3.46·87-s − 1.09·101-s − 0.0153·103-s + 3.18·107-s − 1.88·113-s + 1.02·121-s + 0.000698·127-s − 3.70·129-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.737501205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737501205\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 74 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 622 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1366 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3718 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 59326 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 50230 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 129742 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 452 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 21022 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 414 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 57098 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 422 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 491302 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 586222 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 777358 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 512 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 267770 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1013038 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 714430 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.440770602547108786361063509589, −8.612530794005377364556981245722, −8.345957964689191796166767793417, −7.74275349709763535703654400905, −7.58945692039053130876819925830, −7.02909610025355546473894881567, −6.87121693574072816399190878207, −6.24300115372090712571114381319, −5.76926548997243841302322716052, −5.38125840976665714624537795491, −5.02366455434393441947948880626, −4.21826435798735615537071362669, −4.03967053392608507458100283467, −3.45943586589687769923763457704, −3.42612834916423622981928327301, −2.37397160400869814343777185653, −2.15305536183307778070097210187, −1.86227310477759945181101407425, −1.08935316851399059537173668201, −0.24570329086269576981319465952,
0.24570329086269576981319465952, 1.08935316851399059537173668201, 1.86227310477759945181101407425, 2.15305536183307778070097210187, 2.37397160400869814343777185653, 3.42612834916423622981928327301, 3.45943586589687769923763457704, 4.03967053392608507458100283467, 4.21826435798735615537071362669, 5.02366455434393441947948880626, 5.38125840976665714624537795491, 5.76926548997243841302322716052, 6.24300115372090712571114381319, 6.87121693574072816399190878207, 7.02909610025355546473894881567, 7.58945692039053130876819925830, 7.74275349709763535703654400905, 8.345957964689191796166767793417, 8.612530794005377364556981245722, 9.440770602547108786361063509589
