| L(s) = 1 | + 5-s + 2·13-s − 2·17-s + 25-s + 6·29-s + 4·31-s + 6·37-s − 2·41-s + 8·43-s − 7·49-s − 10·53-s − 12·59-s + 2·61-s + 2·65-s + 4·67-s − 8·71-s + 6·73-s − 8·79-s + 4·83-s − 2·85-s + 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.554·13-s − 0.485·17-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.312·41-s + 1.21·43-s − 49-s − 1.37·53-s − 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s + 0.702·73-s − 0.900·79-s + 0.439·83-s − 0.216·85-s + 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.777037531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.777037531\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.27716508996162, −12.74073540038951, −12.29040151808744, −11.82111111167754, −11.10495492033106, −10.99869012042843, −10.29585207899268, −9.921151317012545, −9.326135321014409, −8.986758145134493, −8.398798113965481, −7.912522061277931, −7.503753551302306, −6.683694284588193, −6.339043697243347, −6.044578670378574, −5.276990003922614, −4.755444291245434, −4.321832818899965, −3.685975583762738, −2.912771309354263, −2.633481962468473, −1.774592228647460, −1.245778168055254, −0.4969293557288790,
0.4969293557288790, 1.245778168055254, 1.774592228647460, 2.633481962468473, 2.912771309354263, 3.685975583762738, 4.321832818899965, 4.755444291245434, 5.276990003922614, 6.044578670378574, 6.339043697243347, 6.683694284588193, 7.503753551302306, 7.912522061277931, 8.398798113965481, 8.986758145134493, 9.326135321014409, 9.921151317012545, 10.29585207899268, 10.99869012042843, 11.10495492033106, 11.82111111167754, 12.29040151808744, 12.74073540038951, 13.27716508996162
