| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 2·13-s − 15-s + 2·17-s − 7·19-s + 21-s + 6·23-s + 25-s − 27-s + 8·29-s − 3·31-s − 35-s + 37-s + 2·39-s − 6·41-s + 45-s − 4·47-s − 6·49-s − 2·51-s + 10·53-s + 7·57-s + 4·59-s − 11·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 1.60·19-s + 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s − 0.538·31-s − 0.169·35-s + 0.164·37-s + 0.320·39-s − 0.937·41-s + 0.149·45-s − 0.583·47-s − 6/7·49-s − 0.280·51-s + 1.37·53-s + 0.927·57-s + 0.520·59-s − 1.40·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−16.50406293780817, −15.93626529586584, −15.11807265856939, −14.87443464565724, −14.16806471229602, −13.45976549120180, −12.95836462211889, −12.47840397024345, −11.97825987906243, −11.22038145462723, −10.68173194502855, −10.11321041360299, −9.722518085639347, −8.831002329551511, −8.472159140594152, −7.511417325955242, −6.895812050040277, −6.380763253403010, −5.823139604279727, −4.949411890305557, −4.626326023231767, −3.613287160804372, −2.837988044522218, −2.038413417958437, −1.074704321022675, 0,
1.074704321022675, 2.038413417958437, 2.837988044522218, 3.613287160804372, 4.626326023231767, 4.949411890305557, 5.823139604279727, 6.380763253403010, 6.895812050040277, 7.511417325955242, 8.472159140594152, 8.831002329551511, 9.722518085639347, 10.11321041360299, 10.68173194502855, 11.22038145462723, 11.97825987906243, 12.47840397024345, 12.95836462211889, 13.45976549120180, 14.16806471229602, 14.87443464565724, 15.11807265856939, 15.93626529586584, 16.50406293780817
