| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 5·11-s − 14-s + 16-s + 6·17-s − 8·19-s + 20-s − 5·22-s − 3·23-s + 25-s − 28-s + 8·29-s − 9·31-s + 32-s + 6·34-s − 35-s + 7·37-s − 8·38-s + 40-s − 7·41-s − 12·43-s − 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.50·11-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.83·19-s + 0.223·20-s − 1.06·22-s − 0.625·23-s + 1/5·25-s − 0.188·28-s + 1.48·29-s − 1.61·31-s + 0.176·32-s + 1.02·34-s − 0.169·35-s + 1.15·37-s − 1.29·38-s + 0.158·40-s − 1.09·41-s − 1.82·43-s − 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.95525348680971, −13.19871483293764, −13.07999453102113, −12.62707643802290, −12.18512673141443, −11.52048402702493, −11.05306855724927, −10.35025005326084, −10.10004742478135, −9.875784758462024, −8.903351729755336, −8.289782120672163, −8.058348413546659, −7.394285020438373, −6.636893148756099, −6.441076641641873, −5.700673671633689, −5.188115916818463, −4.977753423519565, −4.028758082486466, −3.597549992848832, −2.941709386976275, −2.297371597952966, −1.947660163926715, −0.8945466039821658, 0,
0.8945466039821658, 1.947660163926715, 2.297371597952966, 2.941709386976275, 3.597549992848832, 4.028758082486466, 4.977753423519565, 5.188115916818463, 5.700673671633689, 6.441076641641873, 6.636893148756099, 7.394285020438373, 8.058348413546659, 8.289782120672163, 8.903351729755336, 9.875784758462024, 10.10004742478135, 10.35025005326084, 11.05306855724927, 11.52048402702493, 12.18512673141443, 12.62707643802290, 13.07999453102113, 13.19871483293764, 13.95525348680971
