| L(s) = 1 | − 2.35e4·5-s + 5.31e5·9-s − 6.91e6·13-s − 4.72e7·17-s + 3.08e8·25-s + 1.73e8·29-s + 2.05e9·37-s − 2.28e9·41-s − 1.24e10·45-s + 1.38e10·49-s + 4.34e10·53-s + 4.78e10·61-s + 1.62e11·65-s − 1.19e11·73-s + 2.82e11·81-s + 1.11e12·85-s + 9.07e11·89-s + 5.02e11·97-s − 7.77e11·101-s + 3.14e12·109-s − 2.90e12·113-s − 3.67e12·117-s + ⋯ |
| L(s) = 1 | − 1.50·5-s + 9-s − 1.43·13-s − 1.95·17-s + 1.26·25-s + 0.291·29-s + 0.799·37-s − 0.481·41-s − 1.50·45-s + 49-s + 1.96·53-s + 0.928·61-s + 2.15·65-s − 0.791·73-s + 81-s + 2.94·85-s + 1.82·89-s + 0.603·97-s − 0.732·101-s + 1.87·109-s − 1.39·113-s − 1.43·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.014417863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014417863\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 5 | \( 1 + 23506 T + p^{12} T^{2} \) |
| 7 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 11 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 13 | \( 1 + 6911282 T + p^{12} T^{2} \) |
| 17 | \( 1 + 47295038 T + p^{12} T^{2} \) |
| 19 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 23 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( 1 - 173439758 T + p^{12} T^{2} \) |
| 31 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 37 | \( 1 - 2050092718 T + p^{12} T^{2} \) |
| 41 | \( 1 + 2285065118 T + p^{12} T^{2} \) |
| 43 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 47 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 53 | \( 1 - 43462597358 T + p^{12} T^{2} \) |
| 59 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 61 | \( 1 - 47844884878 T + p^{12} T^{2} \) |
| 67 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 71 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 73 | \( 1 + 119852347678 T + p^{12} T^{2} \) |
| 79 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 83 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 89 | \( 1 - 907573615522 T + p^{12} T^{2} \) |
| 97 | \( 1 - 502341690242 T + p^{12} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.28981429367275219544847542069, −11.42481183376254618318339663856, −10.22702584728970049932946222590, −8.850730849914209307852509916236, −7.58467623789627058340656314710, −6.84708991657765450398641837994, −4.74770113374792869350757636604, −3.99211090848928084943702748038, −2.35088883422343064671288788093, −0.52868315052815219983932256660,
0.52868315052815219983932256660, 2.35088883422343064671288788093, 3.99211090848928084943702748038, 4.74770113374792869350757636604, 6.84708991657765450398641837994, 7.58467623789627058340656314710, 8.850730849914209307852509916236, 10.22702584728970049932946222590, 11.42481183376254618318339663856, 12.28981429367275219544847542069
