| L(s) = 1 | + 343·4-s + 1.25e3·5-s − 7.04e3·9-s + 5.08e4·11-s + 5.21e4·16-s + 2.60e5·19-s + 4.28e5·20-s + 1.17e6·25-s − 2.41e6·36-s + 1.74e7·44-s − 8.80e6·45-s + 1.15e7·49-s + 6.35e7·55-s − 4.65e7·61-s − 4.60e6·64-s + 8.94e7·76-s + 6.51e7·80-s + 6.54e6·81-s + 3.25e8·95-s − 3.58e8·99-s + 4.01e8·100-s − 3.91e8·101-s + 1.51e9·121-s + 9.76e8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.33·4-s + 2·5-s − 1.07·9-s + 3.47·11-s + 0.795·16-s + 2·19-s + 2.67·20-s + 3·25-s − 1.43·36-s + 4.65·44-s − 2.14·45-s + 2·49-s + 6.94·55-s − 3.35·61-s − 0.274·64-s + 2.67·76-s + 1.59·80-s + 0.151·81-s + 4·95-s − 3.72·99-s + 4.01·100-s − 3.76·101-s + 7.05·121-s + 4·125-s − 0.853·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(10.37441928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.37441928\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 343 T^{2} + p^{16} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 7042 T^{2} + p^{16} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 25438 T + p^{8} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1137613442 T^{2} + p^{16} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 3455139432962 T^{2} + p^{16} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 105135912979202 T^{2} + p^{16} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 23259362 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 800814673910398 T^{2} + p^{16} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 12005219386199998 T^{2} + p^{16} T^{4} \) |
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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
−12.24958918963094448587899307249, −11.96900630041085929675338776446, −11.80752943138265514005765147707, −10.90816445289999546413956588704, −10.75508121447565835414577087345, −9.661722703716370405830380578792, −9.385287176460921162988514441678, −9.132863576895389497247545246430, −8.405495497754468146662675232937, −7.07321673249459922310926133138, −7.03788453395797962669415934504, −6.16725353261256133144813716080, −6.03105684952951395849527094346, −5.43857660047698064263818172943, −4.35528306719880624987061844726, −3.31641366754077334968475562665, −2.83442074953698338477064371206, −1.95674308547374988458408806991, −1.34036109870488398273920271305, −1.06220855300898680125899061553,
1.06220855300898680125899061553, 1.34036109870488398273920271305, 1.95674308547374988458408806991, 2.83442074953698338477064371206, 3.31641366754077334968475562665, 4.35528306719880624987061844726, 5.43857660047698064263818172943, 6.03105684952951395849527094346, 6.16725353261256133144813716080, 7.03788453395797962669415934504, 7.07321673249459922310926133138, 8.405495497754468146662675232937, 9.132863576895389497247545246430, 9.385287176460921162988514441678, 9.661722703716370405830380578792, 10.75508121447565835414577087345, 10.90816445289999546413956588704, 11.80752943138265514005765147707, 11.96900630041085929675338776446, 12.24958918963094448587899307249
