| L(s) = 1 | + 1.21e3·2-s − 1.52e7·4-s + 3.68e8·5-s − 2.51e10·7-s − 3.90e10·8-s + 2.82e11·9-s + 4.48e11·10-s − 3.05e13·14-s + 2.09e14·16-s + 3.43e14·18-s − 4.07e15·19-s − 5.63e15·20-s + 7.59e16·25-s + 3.84e17·28-s + 7.87e17·31-s + 9.09e17·32-s − 9.25e18·35-s − 4.32e18·36-s − 4.96e18·38-s − 1.43e19·40-s − 1.11e19·41-s + 1.03e20·45-s − 1.69e20·47-s + 4.40e20·49-s + 9.24e19·50-s + 9.81e20·56-s + 7.49e20·59-s + ⋯ |
| L(s) = 1 | + 0.297·2-s − 0.911·4-s + 1.50·5-s − 1.81·7-s − 0.568·8-s + 9-s + 0.448·10-s − 0.539·14-s + 0.742·16-s + 0.297·18-s − 1.84·19-s − 1.37·20-s + 1.27·25-s + 1.65·28-s + 31-s + 0.788·32-s − 2.73·35-s − 0.911·36-s − 0.547·38-s − 0.856·40-s − 0.492·41-s + 1.50·45-s − 1.45·47-s + 2.29·49-s + 0.378·50-s + 1.03·56-s + 0.421·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{25}{2})\) |
\(\approx\) |
\(1.866735950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.866735950\) |
| \(L(13)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 - p^{12} T \) |
| good | 2 | \( 1 - 1217 T + p^{24} T^{2} \) |
| 3 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 5 | \( 1 - 368217506 T + p^{24} T^{2} \) |
| 7 | \( 1 + 25142615998 T + p^{24} T^{2} \) |
| 11 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 13 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 17 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 19 | \( 1 + 4078801295506078 T + p^{24} T^{2} \) |
| 23 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 29 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 37 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 41 | \( 1 + 11103825178190226238 T + p^{24} T^{2} \) |
| 43 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 47 | \( 1 + \)\(16\!\cdots\!18\)\( T + p^{24} T^{2} \) |
| 53 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 59 | \( 1 - \)\(74\!\cdots\!62\)\( T + p^{24} T^{2} \) |
| 61 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 67 | \( 1 - \)\(10\!\cdots\!22\)\( T + p^{24} T^{2} \) |
| 71 | \( 1 + \)\(20\!\cdots\!18\)\( T + p^{24} T^{2} \) |
| 73 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 79 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 83 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 89 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 97 | \( 1 - \)\(11\!\cdots\!82\)\( T + p^{24} T^{2} \) |
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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
−12.71499566823486845034207715282, −10.15149531226698044902650188805, −9.819072069944861188622114217235, −8.759805748665310131061497359347, −6.64473318952669124461905715391, −6.00741573920647408749067347244, −4.61230358816569065263341035789, −3.38339336545768490966850265653, −2.08581288989377390872993297543, −0.60734183116774241876967482246,
0.60734183116774241876967482246, 2.08581288989377390872993297543, 3.38339336545768490966850265653, 4.61230358816569065263341035789, 6.00741573920647408749067347244, 6.64473318952669124461905715391, 8.759805748665310131061497359347, 9.819072069944861188622114217235, 10.15149531226698044902650188805, 12.71499566823486845034207715282
