| L(s) = 1 | + 3·5-s − 7-s − 11-s − 2·13-s + 2·17-s − 6·19-s − 7·23-s + 4·25-s + 8·31-s − 3·35-s + 37-s + 7·41-s + 8·43-s + 13·47-s − 6·49-s − 3·53-s − 3·55-s + 12·59-s + 4·61-s − 6·65-s + 2·67-s + 8·71-s − 7·73-s + 77-s − 6·79-s + 15·83-s + 6·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 0.301·11-s − 0.554·13-s + 0.485·17-s − 1.37·19-s − 1.45·23-s + 4/5·25-s + 1.43·31-s − 0.507·35-s + 0.164·37-s + 1.09·41-s + 1.21·43-s + 1.89·47-s − 6/7·49-s − 0.412·53-s − 0.404·55-s + 1.56·59-s + 0.512·61-s − 0.744·65-s + 0.244·67-s + 0.949·71-s − 0.819·73-s + 0.113·77-s − 0.675·79-s + 1.64·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.788471533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.788471533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.85114188478878, −12.67909512495872, −12.00130539824254, −11.72967344428088, −10.77143830270139, −10.60144671551556, −10.11891039395158, −9.597867222432938, −9.458969064469988, −8.734953341520402, −8.240418726488887, −7.779853862007435, −7.215154289875622, −6.520551948537712, −6.216962359631470, −5.805330181526159, −5.341361043520396, −4.691486688931249, −4.131097913961354, −3.680529660397181, −2.615282608845361, −2.496857960992313, −2.002310732868614, −1.160735633432560, −0.4702293309008313,
0.4702293309008313, 1.160735633432560, 2.002310732868614, 2.496857960992313, 2.615282608845361, 3.680529660397181, 4.131097913961354, 4.691486688931249, 5.341361043520396, 5.805330181526159, 6.216962359631470, 6.520551948537712, 7.215154289875622, 7.779853862007435, 8.240418726488887, 8.734953341520402, 9.458969064469988, 9.597867222432938, 10.11891039395158, 10.60144671551556, 10.77143830270139, 11.72967344428088, 12.00130539824254, 12.67909512495872, 12.85114188478878
