| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s − 7·13-s + 15-s − 3·19-s − 21-s − 9·23-s − 4·25-s − 27-s − 6·29-s + 4·31-s + 33-s − 35-s + 10·37-s + 7·39-s − 3·41-s − 3·43-s − 45-s + 8·47-s + 49-s − 4·53-s + 55-s + 3·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.94·13-s + 0.258·15-s − 0.688·19-s − 0.218·21-s − 1.87·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.169·35-s + 1.64·37-s + 1.12·39-s − 0.468·41-s − 0.457·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.549·53-s + 0.134·55-s + 0.397·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.15057810091459, −13.54333604373219, −12.83275069375530, −12.55230261346457, −11.94729256654220, −11.70133334918939, −11.20699197349664, −10.56299724728515, −10.07538745685662, −9.691830616817017, −9.249538502133209, −8.298262697098297, −7.956846691106286, −7.556731081302192, −7.059226688382994, −6.367941677341805, −5.800024457438384, −5.394509442402854, −4.537847306384002, −4.427102255596869, −3.750753429266309, −2.859376724044774, −2.171267459945435, −1.810281903388643, −0.5865672582315127, 0,
0.5865672582315127, 1.810281903388643, 2.171267459945435, 2.859376724044774, 3.750753429266309, 4.427102255596869, 4.537847306384002, 5.394509442402854, 5.800024457438384, 6.367941677341805, 7.059226688382994, 7.556731081302192, 7.956846691106286, 8.298262697098297, 9.249538502133209, 9.691830616817017, 10.07538745685662, 10.56299724728515, 11.20699197349664, 11.70133334918939, 11.94729256654220, 12.55230261346457, 12.83275069375530, 13.54333604373219, 14.15057810091459
