| L(s) = 1 | − 2·3-s + 7-s + 9-s + 3·11-s + 7·13-s − 3·17-s − 19-s − 2·21-s − 9·23-s + 4·27-s − 9·29-s − 8·31-s − 6·33-s + 4·37-s − 14·39-s − 3·41-s − 4·43-s + 6·47-s + 49-s + 6·51-s + 12·53-s + 2·57-s + 9·59-s − 61-s + 63-s − 67-s + 18·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.94·13-s − 0.727·17-s − 0.229·19-s − 0.436·21-s − 1.87·23-s + 0.769·27-s − 1.67·29-s − 1.43·31-s − 1.04·33-s + 0.657·37-s − 2.24·39-s − 0.468·41-s − 0.609·43-s + 0.875·47-s + 1/7·49-s + 0.840·51-s + 1.64·53-s + 0.264·57-s + 1.17·59-s − 0.128·61-s + 0.125·63-s − 0.122·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.74973825054985, −14.20620814291390, −13.46925307055183, −13.32569854816496, −12.58102105693691, −11.90479049395628, −11.61844217803659, −11.21652228960052, −10.66196477196631, −10.40399200098941, −9.425285236117360, −9.037004580782253, −8.457358955400439, −7.979230582227421, −7.148421653550240, −6.629763343120753, −6.092353559899144, −5.694743976997993, −5.294604916632715, −4.174604248163433, −4.047708744595523, −3.418079153497678, −2.169703325432224, −1.672834242875564, −0.8739409019482574, 0,
0.8739409019482574, 1.672834242875564, 2.169703325432224, 3.418079153497678, 4.047708744595523, 4.174604248163433, 5.294604916632715, 5.694743976997993, 6.092353559899144, 6.629763343120753, 7.148421653550240, 7.979230582227421, 8.457358955400439, 9.037004580782253, 9.425285236117360, 10.40399200098941, 10.66196477196631, 11.21652228960052, 11.61844217803659, 11.90479049395628, 12.58102105693691, 13.32569854816496, 13.46925307055183, 14.20620814291390, 14.74973825054985
