| L(s) = 1 | − 2·5-s + 2·7-s + 13-s − 2·17-s − 4·23-s − 25-s + 2·29-s + 2·31-s − 4·35-s + 6·37-s + 6·41-s + 2·43-s − 6·47-s − 3·49-s − 6·53-s + 8·59-s − 2·61-s − 2·65-s + 8·67-s + 2·71-s − 14·73-s + 4·79-s − 8·83-s + 4·85-s + 14·89-s + 2·91-s + 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 0.277·13-s − 0.485·17-s − 0.834·23-s − 1/5·25-s + 0.371·29-s + 0.359·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s + 0.304·43-s − 0.875·47-s − 3/7·49-s − 0.824·53-s + 1.04·59-s − 0.256·61-s − 0.248·65-s + 0.977·67-s + 0.237·71-s − 1.63·73-s + 0.450·79-s − 0.878·83-s + 0.433·85-s + 1.48·89-s + 0.209·91-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610403191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610403191\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.94321759061702, −15.66353518993258, −14.85850302280320, −14.51335064592668, −13.91702078405092, −13.23078824898547, −12.69095525355953, −11.99892284372783, −11.45092925915429, −11.21561244056841, −10.45635725632709, −9.809400138014201, −9.121685942451723, −8.398637866846042, −7.948108576409562, −7.562522633867154, −6.673609516555069, −6.103090585456340, −5.307788575150000, −4.488605178324011, −4.155704528846821, −3.329441081067830, −2.446616028378223, −1.598293140672586, −0.5623645166211149,
0.5623645166211149, 1.598293140672586, 2.446616028378223, 3.329441081067830, 4.155704528846821, 4.488605178324011, 5.307788575150000, 6.103090585456340, 6.673609516555069, 7.562522633867154, 7.948108576409562, 8.398637866846042, 9.121685942451723, 9.809400138014201, 10.45635725632709, 11.21561244056841, 11.45092925915429, 11.99892284372783, 12.69095525355953, 13.23078824898547, 13.91702078405092, 14.51335064592668, 14.85850302280320, 15.66353518993258, 15.94321759061702
