| L(s) = 1 | + 5-s − 6·9-s − 4·13-s + 25-s + 16·31-s + 12·37-s − 12·41-s + 16·43-s − 6·45-s + 2·49-s + 12·53-s − 4·65-s − 16·67-s + 27·81-s + 32·83-s − 12·89-s + 24·117-s − 6·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·155-s + 157-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 2·9-s − 1.10·13-s + 1/5·25-s + 2.87·31-s + 1.97·37-s − 1.87·41-s + 2.43·43-s − 0.894·45-s + 2/7·49-s + 1.64·53-s − 0.496·65-s − 1.95·67-s + 3·81-s + 3.51·83-s − 1.27·89-s + 2.21·117-s − 0.545·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.288938274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288938274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.291763249756947680273677410722, −8.916494500187803195551794457755, −8.521590142691914902934262227672, −7.78749485529796929229571735762, −7.75327442992794184642667314086, −6.76167065115892628778425003984, −6.34551581674059305006577784340, −5.84906736542976130410760055690, −5.42444342291268605029621774940, −4.76905661119552301505559653794, −4.28670681048457803999561096522, −3.26209478229515851009460820825, −2.60068110583024471076066537498, −2.39961978181641342090490014624, −0.77815604840316859728659905693,
0.77815604840316859728659905693, 2.39961978181641342090490014624, 2.60068110583024471076066537498, 3.26209478229515851009460820825, 4.28670681048457803999561096522, 4.76905661119552301505559653794, 5.42444342291268605029621774940, 5.84906736542976130410760055690, 6.34551581674059305006577784340, 6.76167065115892628778425003984, 7.75327442992794184642667314086, 7.78749485529796929229571735762, 8.521590142691914902934262227672, 8.916494500187803195551794457755, 9.291763249756947680273677410722
