| L(s) = 1 | + 5-s + 4·7-s − 2·11-s − 13-s − 2·17-s − 6·19-s + 6·23-s + 25-s − 2·29-s + 6·31-s + 4·35-s − 2·37-s − 10·41-s + 10·43-s − 12·47-s + 9·49-s − 2·53-s − 2·55-s + 10·59-s + 2·61-s − 65-s + 12·67-s + 10·71-s + 10·73-s − 8·77-s + 4·79-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.603·11-s − 0.277·13-s − 0.485·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s − 0.371·29-s + 1.07·31-s + 0.676·35-s − 0.328·37-s − 1.56·41-s + 1.52·43-s − 1.75·47-s + 9/7·49-s − 0.274·53-s − 0.269·55-s + 1.30·59-s + 0.256·61-s − 0.124·65-s + 1.46·67-s + 1.18·71-s + 1.17·73-s − 0.911·77-s + 0.450·79-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.559242049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.559242049\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.897773872696157958243371649683, −6.93382221399178540379656519347, −6.47403483990558052185808471960, −5.44081773836520664454715949181, −4.94152921655387894445681879278, −4.47475998458279826181189336005, −3.42625638274975134471108613648, −2.32195899884046682848721253631, −1.92875473556257178087803423750, −0.76823693791712766521382629362,
0.76823693791712766521382629362, 1.92875473556257178087803423750, 2.32195899884046682848721253631, 3.42625638274975134471108613648, 4.47475998458279826181189336005, 4.94152921655387894445681879278, 5.44081773836520664454715949181, 6.47403483990558052185808471960, 6.93382221399178540379656519347, 7.897773872696157958243371649683
