| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 4·11-s + 14-s + 16-s − 5·17-s + 2·19-s + 20-s − 4·22-s + 4·23-s + 25-s + 28-s + 5·29-s − 2·31-s + 32-s − 5·34-s + 35-s − 8·37-s + 2·38-s + 40-s + 6·43-s − 4·44-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.267·14-s + 1/4·16-s − 1.21·17-s + 0.458·19-s + 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s + 0.188·28-s + 0.928·29-s − 0.359·31-s + 0.176·32-s − 0.857·34-s + 0.169·35-s − 1.31·37-s + 0.324·38-s + 0.158·40-s + 0.914·43-s − 0.603·44-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.78878122035379, −13.57784017104243, −12.91443419553706, −12.65156728997905, −12.14472083619902, −11.37754780151020, −11.00079795575866, −10.77422632115583, −9.974890907517646, −9.716668030757184, −8.857191797927575, −8.513162966053908, −7.896578792010050, −7.380054252877846, −6.692816475471678, −6.509577623086765, −5.573698113098088, −5.273640143014003, −4.793071488673035, −4.309175796841217, −3.454167532294880, −2.951249821491219, −2.335979106688998, −1.841003904789661, −1.000119867709962, 0,
1.000119867709962, 1.841003904789661, 2.335979106688998, 2.951249821491219, 3.454167532294880, 4.309175796841217, 4.793071488673035, 5.273640143014003, 5.573698113098088, 6.509577623086765, 6.692816475471678, 7.380054252877846, 7.896578792010050, 8.513162966053908, 8.857191797927575, 9.716668030757184, 9.974890907517646, 10.77422632115583, 11.00079795575866, 11.37754780151020, 12.14472083619902, 12.65156728997905, 12.91443419553706, 13.57784017104243, 13.78878122035379
