| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 6·13-s + 15-s + 3·17-s − 6·19-s + 21-s + 23-s − 4·25-s − 27-s − 29-s − 4·31-s + 35-s + 37-s − 6·39-s + 2·41-s + 8·43-s − 45-s + 4·47-s − 6·49-s − 3·51-s + 4·53-s + 6·57-s − 12·59-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.258·15-s + 0.727·17-s − 1.37·19-s + 0.218·21-s + 0.208·23-s − 4/5·25-s − 0.192·27-s − 0.185·29-s − 0.718·31-s + 0.169·35-s + 0.164·37-s − 0.960·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s + 0.583·47-s − 6/7·49-s − 0.420·51-s + 0.549·53-s + 0.794·57-s − 1.56·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.12863497131443, −12.76075371266794, −12.24308787997407, −11.92104379935908, −11.16410419331973, −10.98843190267046, −10.60405043262320, −10.04570834889212, −9.472207056607928, −8.871804194339057, −8.666798326457198, −7.881132313574859, −7.570419650190992, −7.052696547749947, −6.239287204626091, −6.057164326240337, −5.765186348455895, −4.917342241227238, −4.294771846857147, −3.992987510272013, −3.378324562685887, −2.901261423238078, −1.942585358273212, −1.467048084864620, −0.7019456171608123, 0,
0.7019456171608123, 1.467048084864620, 1.942585358273212, 2.901261423238078, 3.378324562685887, 3.992987510272013, 4.294771846857147, 4.917342241227238, 5.765186348455895, 6.057164326240337, 6.239287204626091, 7.052696547749947, 7.570419650190992, 7.881132313574859, 8.666798326457198, 8.871804194339057, 9.472207056607928, 10.04570834889212, 10.60405043262320, 10.98843190267046, 11.16410419331973, 11.92104379935908, 12.24308787997407, 12.76075371266794, 13.12863497131443
