| L(s) = 1 | + 3-s − 5-s + 3·7-s − 2·9-s − 11-s − 15-s + 17-s + 19-s + 3·21-s + 5·23-s + 25-s − 5·27-s + 3·29-s − 8·31-s − 33-s − 3·35-s + 9·37-s − 5·41-s − 9·43-s + 2·45-s − 8·47-s + 2·49-s + 51-s − 10·53-s + 55-s + 57-s − 3·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s − 0.301·11-s − 0.258·15-s + 0.242·17-s + 0.229·19-s + 0.654·21-s + 1.04·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s − 1.43·31-s − 0.174·33-s − 0.507·35-s + 1.47·37-s − 0.780·41-s − 1.37·43-s + 0.298·45-s − 1.16·47-s + 2/7·49-s + 0.140·51-s − 1.37·53-s + 0.134·55-s + 0.132·57-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−16.58727234897369, −15.85844101782024, −15.05024267073156, −14.88761257173031, −14.33150707183830, −13.84071816456701, −13.06725237495859, −12.69231399523790, −11.61859771342378, −11.51157114069622, −10.94954356700390, −10.19106555650271, −9.482640324043486, −8.739557637559049, −8.438737805032268, −7.684669446487089, −7.446165642663951, −6.442562272180687, −5.694358028846517, −4.907642802541237, −4.577063121024682, −3.395651011726596, −3.085644761654025, −2.065017869857931, −1.304038468395442, 0,
1.304038468395442, 2.065017869857931, 3.085644761654025, 3.395651011726596, 4.577063121024682, 4.907642802541237, 5.694358028846517, 6.442562272180687, 7.446165642663951, 7.684669446487089, 8.438737805032268, 8.739557637559049, 9.482640324043486, 10.19106555650271, 10.94954356700390, 11.51157114069622, 11.61859771342378, 12.69231399523790, 13.06725237495859, 13.84071816456701, 14.33150707183830, 14.88761257173031, 15.05024267073156, 15.85844101782024, 16.58727234897369
