| L(s) = 1 | + 38·7-s + 604·13-s + 608·19-s + 161·25-s − 478·31-s + 1.48e3·37-s + 1.96e3·43-s − 3.71e3·49-s − 632·61-s − 9.24e3·67-s − 6.06e3·73-s + 2.09e4·79-s + 2.29e4·91-s − 1.30e4·97-s + 1.53e4·103-s + 4.64e3·109-s + 1.41e4·121-s + 127-s + 131-s + 2.31e4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.775·7-s + 3.57·13-s + 1.68·19-s + 0.257·25-s − 0.497·31-s + 1.08·37-s + 1.06·43-s − 1.54·49-s − 0.169·61-s − 2.05·67-s − 1.13·73-s + 3.34·79-s + 2.77·91-s − 1.38·97-s + 1.44·103-s + 0.391·109-s + 0.966·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 1.30·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.233539785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.233539785\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 - 161 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 19 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 14153 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 302 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4354 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 16 p T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 469682 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 954878 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 239 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 20 p T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5599538 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 982 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5067806 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 13243313 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 15696638 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 316 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4622 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 47518238 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3031 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10450 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 64676047 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 76456478 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6517 T + p^{4} T^{2} )^{2} \) |
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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
−11.01291695289338665195092322068, −10.42268196153872591871060427051, −9.899310952965056480293515345853, −9.102404334340660728381761442422, −9.087360742199890235217641431341, −8.509852563490533973890280676430, −7.938443686123466830008390377042, −7.77244114411234898435382920482, −7.10318271440601160754626563773, −6.24307762093820969345085424808, −6.19787216911581109782498043429, −5.61916975788905037051490954415, −5.01502644700367161956130388076, −4.41178519547947471568620883523, −3.68561599692514857801255680692, −3.46903325266830113036909626391, −2.72057606706938499083571241025, −1.58674060797695215265115305654, −1.30723021924889872477913291184, −0.71613391535393847887768168932,
0.71613391535393847887768168932, 1.30723021924889872477913291184, 1.58674060797695215265115305654, 2.72057606706938499083571241025, 3.46903325266830113036909626391, 3.68561599692514857801255680692, 4.41178519547947471568620883523, 5.01502644700367161956130388076, 5.61916975788905037051490954415, 6.19787216911581109782498043429, 6.24307762093820969345085424808, 7.10318271440601160754626563773, 7.77244114411234898435382920482, 7.938443686123466830008390377042, 8.509852563490533973890280676430, 9.087360742199890235217641431341, 9.102404334340660728381761442422, 9.899310952965056480293515345853, 10.42268196153872591871060427051, 11.01291695289338665195092322068
