| L(s) = 1 | − 6.55e4·2-s + 4.29e9·4-s + 1.96e11·5-s − 2.81e14·8-s − 1.28e16·10-s + 1.33e18·13-s + 1.84e19·16-s − 1.42e18·17-s + 8.43e20·20-s + 1.53e22·25-s − 8.71e22·26-s − 4.62e23·29-s − 1.20e24·32-s + 9.35e22·34-s + 1.33e25·37-s − 5.53e25·40-s + 1.17e26·41-s + 1.10e27·49-s − 1.00e27·50-s + 5.71e27·52-s + 6.73e27·53-s + 3.03e28·58-s − 7.13e28·61-s + 7.92e28·64-s + 2.61e29·65-s − 6.12e27·68-s + 6.06e29·73-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 1.28·5-s − 8-s − 1.28·10-s + 1.99·13-s + 16-s − 0.0293·17-s + 1.28·20-s + 0.658·25-s − 1.99·26-s − 1.84·29-s − 32-s + 0.0293·34-s + 1.08·37-s − 1.28·40-s + 1.84·41-s + 49-s − 0.658·50-s + 1.99·52-s + 1.73·53-s + 1.84·58-s − 1.94·61-s + 64-s + 2.57·65-s − 0.0293·68-s + 0.932·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(2.385203334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.385203334\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{16} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 196496109694 T + p^{32} T^{2} \) |
| 7 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 11 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 13 | \( 1 - 1330087744899070082 T + p^{32} T^{2} \) |
| 17 | \( 1 + 1427124567881986562 T + p^{32} T^{2} \) |
| 19 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 23 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 29 | \( 1 + \)\(46\!\cdots\!42\)\( T + p^{32} T^{2} \) |
| 31 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 37 | \( 1 - \)\(13\!\cdots\!82\)\( T + p^{32} T^{2} \) |
| 41 | \( 1 - \)\(11\!\cdots\!18\)\( T + p^{32} T^{2} \) |
| 43 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 47 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 53 | \( 1 - \)\(67\!\cdots\!58\)\( T + p^{32} T^{2} \) |
| 59 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 61 | \( 1 + \)\(71\!\cdots\!78\)\( T + p^{32} T^{2} \) |
| 67 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 71 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 73 | \( 1 - \)\(60\!\cdots\!22\)\( T + p^{32} T^{2} \) |
| 79 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 83 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 89 | \( 1 + \)\(17\!\cdots\!22\)\( T + p^{32} T^{2} \) |
| 97 | \( 1 - \)\(84\!\cdots\!42\)\( T + p^{32} T^{2} \) |
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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
−10.57495822045041636150130912182, −9.449718761555876203750012565430, −8.785758890736362004392887931139, −7.52294237302122104779534004791, −6.16664593226891213239127779972, −5.75096830587602851678865686579, −3.77510695724575015003575820908, −2.47066756082772872700925171594, −1.57554042094668081782344285122, −0.77466875495191367627480584890,
0.77466875495191367627480584890, 1.57554042094668081782344285122, 2.47066756082772872700925171594, 3.77510695724575015003575820908, 5.75096830587602851678865686579, 6.16664593226891213239127779972, 7.52294237302122104779534004791, 8.785758890736362004392887931139, 9.449718761555876203750012565430, 10.57495822045041636150130912182
