L(s) = 1 | − 3-s + 4·5-s − 2·9-s + 3·11-s + 5·13-s − 4·15-s − 17-s − 6·19-s − 4·23-s + 11·25-s + 5·27-s − 8·29-s − 3·33-s + 8·37-s − 5·39-s − 8·41-s − 10·43-s − 8·45-s − 2·47-s + 51-s − 3·53-s + 12·55-s + 6·57-s − 2·59-s + 8·61-s + 20·65-s − 14·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78·5-s − 2/3·9-s + 0.904·11-s + 1.38·13-s − 1.03·15-s − 0.242·17-s − 1.37·19-s − 0.834·23-s + 11/5·25-s + 0.962·27-s − 1.48·29-s − 0.522·33-s + 1.31·37-s − 0.800·39-s − 1.24·41-s − 1.52·43-s − 1.19·45-s − 0.291·47-s + 0.140·51-s − 0.412·53-s + 1.61·55-s + 0.794·57-s − 0.260·59-s + 1.02·61-s + 2.48·65-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.61782797106948, −14.21170920252842, −13.53230522644998, −13.25746473755674, −12.88563417481234, −12.09020429319721, −11.52334866161896, −11.12138605610018, −10.59005766063634, −10.11997369239722, −9.542425440355935, −8.995929421060638, −8.608306720116124, −8.111896257529636, −7.051328401186364, −6.396858931473881, −6.183690507192599, −5.871454144885252, −5.181773837877558, −4.554360123512507, −3.765321086840835, −3.154852083660407, −2.175685724518195, −1.804442999154711, −1.104045002630146, 0,
1.104045002630146, 1.804442999154711, 2.175685724518195, 3.154852083660407, 3.765321086840835, 4.554360123512507, 5.181773837877558, 5.871454144885252, 6.183690507192599, 6.396858931473881, 7.051328401186364, 8.111896257529636, 8.608306720116124, 8.995929421060638, 9.542425440355935, 10.11997369239722, 10.59005766063634, 11.12138605610018, 11.52334866161896, 12.09020429319721, 12.88563417481234, 13.25746473755674, 13.53230522644998, 14.21170920252842, 14.61782797106948