| L(s) = 1 | − 2·5-s − 7-s + 6·11-s + 7·13-s − 2·17-s − 25-s + 10·29-s − 8·31-s + 2·35-s − 7·37-s + 9·43-s − 8·47-s − 6·49-s + 2·53-s − 12·55-s − 4·59-s − 7·61-s − 14·65-s + 3·67-s − 12·71-s − 14·73-s − 6·77-s + 8·79-s − 14·83-s + 4·85-s − 12·89-s − 7·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.80·11-s + 1.94·13-s − 0.485·17-s − 1/5·25-s + 1.85·29-s − 1.43·31-s + 0.338·35-s − 1.15·37-s + 1.37·43-s − 1.16·47-s − 6/7·49-s + 0.274·53-s − 1.61·55-s − 0.520·59-s − 0.896·61-s − 1.73·65-s + 0.366·67-s − 1.42·71-s − 1.63·73-s − 0.683·77-s + 0.900·79-s − 1.53·83-s + 0.433·85-s − 1.27·89-s − 0.733·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.81138831162514, −12.38522170910151, −11.93743904700754, −11.49659369247931, −11.21038305674800, −10.69550885671684, −10.29686223517264, −9.509710971097910, −9.155261105451866, −8.779786901150932, −8.307929424121228, −7.979544548163905, −7.130343724075291, −6.817894353875104, −6.449996591997688, −5.888549410490726, −5.521484682156163, −4.442493257760395, −4.274434629078386, −3.852250033793036, −3.193495955316645, −3.016909640565466, −1.712374805697209, −1.528802404218628, −0.7914796659154026, 0,
0.7914796659154026, 1.528802404218628, 1.712374805697209, 3.016909640565466, 3.193495955316645, 3.852250033793036, 4.274434629078386, 4.442493257760395, 5.521484682156163, 5.888549410490726, 6.449996591997688, 6.817894353875104, 7.130343724075291, 7.979544548163905, 8.307929424121228, 8.779786901150932, 9.155261105451866, 9.509710971097910, 10.29686223517264, 10.69550885671684, 11.21038305674800, 11.49659369247931, 11.93743904700754, 12.38522170910151, 12.81138831162514
