| L(s) = 1 | + 3-s + 7-s + 9-s + 11-s − 2·13-s − 6·17-s + 4·19-s + 21-s + 27-s + 6·29-s − 8·31-s + 33-s + 10·37-s − 2·39-s − 6·41-s + 8·43-s + 49-s − 6·51-s − 6·53-s + 4·57-s − 12·59-s + 2·61-s + 63-s − 4·67-s − 12·71-s + 10·73-s + 77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.218·21-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s + 1.64·37-s − 0.320·39-s − 0.937·41-s + 1.21·43-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 0.256·61-s + 0.125·63-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.99691543596842, −13.77327897461753, −13.07838210905210, −12.72399270988011, −12.12541276439231, −11.62786205386885, −11.09051013265856, −10.72200402003888, −10.02994268066897, −9.499212731090237, −9.100500168002802, −8.684389483074310, −8.021134602983636, −7.530831772318947, −7.152473681712552, −6.464922226948986, −5.996706997113230, −5.231277745339017, −4.612694768793089, −4.324682725159243, −3.539421316437680, −2.909164240423841, −2.355596602319647, −1.720814630312896, −0.9960331115817504, 0,
0.9960331115817504, 1.720814630312896, 2.355596602319647, 2.909164240423841, 3.539421316437680, 4.324682725159243, 4.612694768793089, 5.231277745339017, 5.996706997113230, 6.464922226948986, 7.152473681712552, 7.530831772318947, 8.021134602983636, 8.684389483074310, 9.100500168002802, 9.499212731090237, 10.02994268066897, 10.72200402003888, 11.09051013265856, 11.62786205386885, 12.12541276439231, 12.72399270988011, 13.07838210905210, 13.77327897461753, 13.99691543596842
