| L(s) = 1 | − 3-s + 5-s + 9-s + 2·13-s − 15-s + 4·19-s + 8·23-s + 25-s − 27-s − 2·29-s + 4·31-s + 6·37-s − 2·39-s + 6·41-s − 8·43-s + 45-s + 12·47-s − 7·49-s − 6·53-s − 4·57-s + 12·59-s − 2·61-s + 2·65-s + 16·67-s − 8·69-s + 12·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s + 0.149·45-s + 1.75·47-s − 49-s − 0.824·53-s − 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 1.95·67-s − 0.963·69-s + 1.42·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.497176172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.497176172\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.78768961390035, −12.26825833362210, −11.85733918306466, −11.20478285639146, −10.99837765715080, −10.66702360668137, −9.841206936647921, −9.564625606243041, −9.291285881822084, −8.481041201171027, −8.204088961286371, −7.501771780319339, −7.090571851215860, −6.488932071617755, −6.264592313702651, −5.531004789999413, −5.125619216638633, −4.863190628171817, −3.958656827259140, −3.664438678705688, −2.832515799915329, −2.482343255954211, −1.618282607360409, −1.003530966150411, −0.6329266996351931,
0.6329266996351931, 1.003530966150411, 1.618282607360409, 2.482343255954211, 2.832515799915329, 3.664438678705688, 3.958656827259140, 4.863190628171817, 5.125619216638633, 5.531004789999413, 6.264592313702651, 6.488932071617755, 7.090571851215860, 7.501771780319339, 8.204088961286371, 8.481041201171027, 9.291285881822084, 9.564625606243041, 9.841206936647921, 10.66702360668137, 10.99837765715080, 11.20478285639146, 11.85733918306466, 12.26825833362210, 12.78768961390035
