| L(s) = 1 | + 7-s − 5·13-s − 19-s + 6·23-s + 5·31-s + 37-s − 5·43-s − 12·47-s − 6·49-s + 12·53-s − 12·59-s − 61-s + 13·67-s − 6·71-s − 2·73-s + 8·79-s − 6·83-s + 6·89-s − 5·91-s − 17·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.38·13-s − 0.229·19-s + 1.25·23-s + 0.898·31-s + 0.164·37-s − 0.762·43-s − 1.75·47-s − 6/7·49-s + 1.64·53-s − 1.56·59-s − 0.128·61-s + 1.58·67-s − 0.712·71-s − 0.234·73-s + 0.900·79-s − 0.658·83-s + 0.635·89-s − 0.524·91-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.665171279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665171279\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−12.81926582943300, −12.31080690858135, −11.90769213930062, −11.47574911581333, −10.99105604305951, −10.53652595409320, −9.968239593598368, −9.618881364940321, −9.201570978752187, −8.494012327620334, −8.207042612836243, −7.653470044753799, −7.156509746408248, −6.694258062065522, −6.309137849627072, −5.484380406335404, −5.107262062909761, −4.673486321788742, −4.257768457146044, −3.420972795805963, −2.954861240787036, −2.427463268368506, −1.792394753673858, −1.162609513960620, −0.3611595683722777,
0.3611595683722777, 1.162609513960620, 1.792394753673858, 2.427463268368506, 2.954861240787036, 3.420972795805963, 4.257768457146044, 4.673486321788742, 5.107262062909761, 5.484380406335404, 6.309137849627072, 6.694258062065522, 7.156509746408248, 7.653470044753799, 8.207042612836243, 8.494012327620334, 9.201570978752187, 9.618881364940321, 9.968239593598368, 10.53652595409320, 10.99105604305951, 11.47574911581333, 11.90769213930062, 12.31080690858135, 12.81926582943300
