| L(s) = 1 | − 4·5-s + 2·7-s − 2·11-s − 6·17-s − 2·19-s − 8·23-s + 11·25-s + 6·29-s − 10·31-s − 8·35-s + 4·37-s + 4·43-s − 2·47-s − 3·49-s − 10·53-s + 8·55-s + 14·59-s + 2·61-s − 2·67-s − 6·71-s − 8·73-s − 4·77-s − 6·83-s + 24·85-s + 8·95-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s − 0.603·11-s − 1.45·17-s − 0.458·19-s − 1.66·23-s + 11/5·25-s + 1.11·29-s − 1.79·31-s − 1.35·35-s + 0.657·37-s + 0.609·43-s − 0.291·47-s − 3/7·49-s − 1.37·53-s + 1.07·55-s + 1.82·59-s + 0.256·61-s − 0.244·67-s − 0.712·71-s − 0.936·73-s − 0.455·77-s − 0.658·83-s + 2.60·85-s + 0.820·95-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−14.36720138730817, −13.89743381627162, −13.12479187888287, −12.71986049875162, −12.36152833564506, −11.60669038444496, −11.37298875777171, −11.07666260070472, −10.42102692203163, −10.01046939856385, −9.047783278323299, −8.712561904662169, −8.163987529693474, −7.847058683013834, −7.390643946895460, −6.809760219041196, −6.234440248420488, −5.522401525178273, −4.755243512266143, −4.470368092740871, −3.963522679552262, −3.439818346158049, −2.586536237875075, −2.073271696707477, −1.184825225248267, 0, 0,
1.184825225248267, 2.073271696707477, 2.586536237875075, 3.439818346158049, 3.963522679552262, 4.470368092740871, 4.755243512266143, 5.522401525178273, 6.234440248420488, 6.809760219041196, 7.390643946895460, 7.847058683013834, 8.163987529693474, 8.712561904662169, 9.047783278323299, 10.01046939856385, 10.42102692203163, 11.07666260070472, 11.37298875777171, 11.60669038444496, 12.36152833564506, 12.71986049875162, 13.12479187888287, 13.89743381627162, 14.36720138730817
