| L(s) = 1 | − 512·4-s + 1.25e4·7-s − 1.18e5·13-s + 2.62e5·16-s − 9.76e5·19-s − 6.44e6·28-s + 1.69e6·31-s + 1.53e7·37-s + 1.65e7·43-s + 1.17e8·49-s + 6.06e7·52-s − 1.17e8·61-s − 1.34e8·64-s − 1.12e8·67-s − 2.96e8·73-s + 5.00e8·76-s − 6.16e8·79-s − 1.48e9·91-s − 1.28e9·97-s − 6.22e7·103-s + 2.24e9·109-s + 3.29e9·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.98·7-s − 1.14·13-s + 16-s − 1.71·19-s − 1.98·28-s + 0.328·31-s + 1.34·37-s + 0.739·43-s + 2.92·49-s + 1.14·52-s − 1.09·61-s − 64-s − 0.682·67-s − 1.22·73-s + 1.71·76-s − 1.78·79-s − 2.27·91-s − 1.47·97-s − 0.0545·103-s + 1.52·109-s + 1.98·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + p^{9} T^{2} \) |
| 7 | \( 1 - 12580 T + p^{9} T^{2} \) |
| 11 | \( 1 + p^{9} T^{2} \) |
| 13 | \( 1 + 118370 T + p^{9} T^{2} \) |
| 17 | \( 1 + p^{9} T^{2} \) |
| 19 | \( 1 + 976696 T + p^{9} T^{2} \) |
| 23 | \( 1 + p^{9} T^{2} \) |
| 29 | \( 1 + p^{9} T^{2} \) |
| 31 | \( 1 - 1691228 T + p^{9} T^{2} \) |
| 37 | \( 1 - 15384490 T + p^{9} T^{2} \) |
| 41 | \( 1 + p^{9} T^{2} \) |
| 43 | \( 1 - 16577080 T + p^{9} T^{2} \) |
| 47 | \( 1 + p^{9} T^{2} \) |
| 53 | \( 1 + p^{9} T^{2} \) |
| 59 | \( 1 + p^{9} T^{2} \) |
| 61 | \( 1 + 117903058 T + p^{9} T^{2} \) |
| 67 | \( 1 + 112542320 T + p^{9} T^{2} \) |
| 71 | \( 1 + p^{9} T^{2} \) |
| 73 | \( 1 + 296368310 T + p^{9} T^{2} \) |
| 79 | \( 1 + 616732324 T + p^{9} T^{2} \) |
| 83 | \( 1 + p^{9} T^{2} \) |
| 89 | \( 1 + p^{9} T^{2} \) |
| 97 | \( 1 + 1288928270 T + p^{9} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.16620840367425911579651403739, −8.977040876502007905648246241213, −8.223478399290433145397497572053, −7.45864063107296110658187719632, −5.79339326847068963717658293864, −4.67587495658612626627992611135, −4.31908236884405277372124210304, −2.42724950081479469228731187016, −1.28635390640511919913836873190, 0,
1.28635390640511919913836873190, 2.42724950081479469228731187016, 4.31908236884405277372124210304, 4.67587495658612626627992611135, 5.79339326847068963717658293864, 7.45864063107296110658187719632, 8.223478399290433145397497572053, 8.977040876502007905648246241213, 10.16620840367425911579651403739
