| L(s) = 1 | − 3·5-s − 5·7-s − 6·11-s + 4·13-s − 3·17-s − 19-s + 4·25-s − 29-s + 4·31-s + 15·35-s + 37-s + 9·41-s − 7·43-s − 3·47-s + 18·49-s − 6·53-s + 18·55-s − 3·59-s + 10·61-s − 12·65-s − 4·67-s + 12·71-s + 2·73-s + 30·77-s − 14·79-s + 9·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.88·7-s − 1.80·11-s + 1.10·13-s − 0.727·17-s − 0.229·19-s + 4/5·25-s − 0.185·29-s + 0.718·31-s + 2.53·35-s + 0.164·37-s + 1.40·41-s − 1.06·43-s − 0.437·47-s + 18/7·49-s − 0.824·53-s + 2.42·55-s − 0.390·59-s + 1.28·61-s − 1.48·65-s − 0.488·67-s + 1.42·71-s + 0.234·73-s + 3.41·77-s − 1.57·79-s + 0.976·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.98459138261796, −15.67446728620061, −15.50228567167359, −14.70931509762117, −13.71956601130837, −13.27921775013533, −12.84850383090986, −12.51541965491756, −11.68589747254654, −11.10159570348837, −10.63709865961803, −10.01489258136613, −9.452822093671317, −8.613044503628425, −8.236919263938541, −7.554695231146666, −6.979299146796088, −6.299737923903504, −5.814161435497967, −4.867740813366091, −4.160872590643359, −3.470115365803785, −3.034882452940357, −2.274729413169568, −0.6799666439785377, 0,
0.6799666439785377, 2.274729413169568, 3.034882452940357, 3.470115365803785, 4.160872590643359, 4.867740813366091, 5.814161435497967, 6.299737923903504, 6.979299146796088, 7.554695231146666, 8.236919263938541, 8.613044503628425, 9.452822093671317, 10.01489258136613, 10.63709865961803, 11.10159570348837, 11.68589747254654, 12.51541965491756, 12.84850383090986, 13.27921775013533, 13.71956601130837, 14.70931509762117, 15.50228567167359, 15.67446728620061, 15.98459138261796
