| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 2·7-s − 2·9-s + 11-s + 2·12-s − 4·13-s + 4·14-s − 4·16-s + 2·17-s − 4·18-s + 2·21-s + 2·22-s + 23-s − 8·26-s − 5·27-s + 4·28-s + 7·31-s − 8·32-s + 33-s + 4·34-s − 4·36-s − 3·37-s − 4·39-s − 8·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 0.755·7-s − 2/3·9-s + 0.301·11-s + 0.577·12-s − 1.10·13-s + 1.06·14-s − 16-s + 0.485·17-s − 0.942·18-s + 0.436·21-s + 0.426·22-s + 0.208·23-s − 1.56·26-s − 0.962·27-s + 0.755·28-s + 1.25·31-s − 1.41·32-s + 0.174·33-s + 0.685·34-s − 2/3·36-s − 0.493·37-s − 0.640·39-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.838038282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.838038282\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−11.91373644505446088258662904950, −11.50442818083742402892485916796, −10.07566606066947823589561871516, −8.906461952934258973434563843342, −7.952532350978681778925001629133, −6.74210091587050441798111318378, −5.49322091026488161466653442724, −4.70136976599285716284507090528, −3.45386803680567664069181238677, −2.35074848930377289705338533775,
2.35074848930377289705338533775, 3.45386803680567664069181238677, 4.70136976599285716284507090528, 5.49322091026488161466653442724, 6.74210091587050441798111318378, 7.952532350978681778925001629133, 8.906461952934258973434563843342, 10.07566606066947823589561871516, 11.50442818083742402892485916796, 11.91373644505446088258662904950
