L(s) = 1 | + 2·2-s − 3·3-s + 3·4-s − 6·6-s + 5·7-s + 4·8-s + 6·9-s − 9·12-s − 2·13-s + 10·14-s + 5·16-s − 6·17-s + 12·18-s − 4·19-s − 15·21-s − 12·24-s + 7·25-s − 4·26-s − 9·27-s + 15·28-s + 16·31-s + 6·32-s − 12·34-s + 18·36-s − 8·38-s + 6·39-s − 30·42-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1.73·3-s + 3/2·4-s − 2.44·6-s + 1.88·7-s + 1.41·8-s + 2·9-s − 2.59·12-s − 0.554·13-s + 2.67·14-s + 5/4·16-s − 1.45·17-s + 2.82·18-s − 0.917·19-s − 3.27·21-s − 2.44·24-s + 7/5·25-s − 0.784·26-s − 1.73·27-s + 2.83·28-s + 2.87·31-s + 1.06·32-s − 2.05·34-s + 3·36-s − 1.29·38-s + 0.960·39-s − 4.62·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.064127008\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.064127008\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.12527416843747569924736155900, −10.81819987192700384126653738571, −10.51081536822856523012362329335, −10.08642791535168582302754099414, −9.446308119477983334854180644566, −8.488621579908768185943012188625, −8.300422109758970489867340675834, −7.85036162334922702114329437952, −6.89889442190176206024077455638, −6.77748434292119410393062732674, −6.50379392837217768899571940823, −5.83743061463752645412166003067, −5.19261235971330291783398588582, −4.85767348395588746651493645720, −4.62146132573888650910578520334, −4.37952033965708351220728726435, −3.42847535983241504989019650171, −2.39770062202245478712646422186, −1.91088873287514032072613703014, −0.956148636789177514059356906960,
0.956148636789177514059356906960, 1.91088873287514032072613703014, 2.39770062202245478712646422186, 3.42847535983241504989019650171, 4.37952033965708351220728726435, 4.62146132573888650910578520334, 4.85767348395588746651493645720, 5.19261235971330291783398588582, 5.83743061463752645412166003067, 6.50379392837217768899571940823, 6.77748434292119410393062732674, 6.89889442190176206024077455638, 7.85036162334922702114329437952, 8.300422109758970489867340675834, 8.488621579908768185943012188625, 9.446308119477983334854180644566, 10.08642791535168582302754099414, 10.51081536822856523012362329335, 10.81819987192700384126653738571, 11.12527416843747569924736155900