| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s − 11-s − 2·13-s − 15-s + 6·19-s + 2·21-s − 6·23-s + 25-s − 27-s − 2·31-s + 33-s − 2·35-s + 6·37-s + 2·39-s + 5·41-s + 5·43-s + 45-s − 4·47-s − 3·49-s − 12·53-s − 55-s − 6·57-s + 8·59-s + 7·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.258·15-s + 1.37·19-s + 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.359·31-s + 0.174·33-s − 0.338·35-s + 0.986·37-s + 0.320·39-s + 0.780·41-s + 0.762·43-s + 0.149·45-s − 0.583·47-s − 3/7·49-s − 1.64·53-s − 0.134·55-s − 0.794·57-s + 1.04·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.70713491433103, −12.38002957010339, −11.69870751324071, −11.39523204795269, −10.99890050870854, −10.27910247213492, −9.968212356742992, −9.612803650501292, −9.348593799578168, −8.662328079230063, −7.930433966048604, −7.748607088053667, −7.056843766749627, −6.759177086528685, −6.069218441995290, −5.775033473260837, −5.421180402598767, −4.686739827813867, −4.400124566604819, −3.587789153962955, −3.209419727027077, −2.540128890766899, −2.071537254430544, −1.323987943167308, −0.6753689816773595, 0,
0.6753689816773595, 1.323987943167308, 2.071537254430544, 2.540128890766899, 3.209419727027077, 3.587789153962955, 4.400124566604819, 4.686739827813867, 5.421180402598767, 5.775033473260837, 6.069218441995290, 6.759177086528685, 7.056843766749627, 7.748607088053667, 7.930433966048604, 8.662328079230063, 9.348593799578168, 9.612803650501292, 9.968212356742992, 10.27910247213492, 10.99890050870854, 11.39523204795269, 11.69870751324071, 12.38002957010339, 12.70713491433103
