| L(s) = 1 | − 2-s + 4-s − 3·5-s − 2·7-s − 8-s + 3·10-s + 11-s + 2·14-s + 16-s − 17-s + 6·19-s − 3·20-s − 22-s − 8·23-s + 4·25-s − 2·28-s − 29-s − 32-s + 34-s + 6·35-s + 3·37-s − 6·38-s + 3·40-s + 11·41-s + 4·43-s + 44-s + 8·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.755·7-s − 0.353·8-s + 0.948·10-s + 0.301·11-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 1.37·19-s − 0.670·20-s − 0.213·22-s − 1.66·23-s + 4/5·25-s − 0.377·28-s − 0.185·29-s − 0.176·32-s + 0.171·34-s + 1.01·35-s + 0.493·37-s − 0.973·38-s + 0.474·40-s + 1.71·41-s + 0.609·43-s + 0.150·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−15.62345916144850, −14.69118508054562, −14.44570906848090, −13.73815253729166, −12.93114185531709, −12.59859264958217, −11.86920607716773, −11.54742031107410, −11.19862914536143, −10.37269288002303, −9.801596439125486, −9.469385109945099, −8.758423711071202, −8.195611089831604, −7.532405990079204, −7.448632514826294, −6.543475331438847, −6.069257871926116, −5.371934418287358, −4.362476984906328, −3.960591507626830, −3.256116455285572, −2.703487377077499, −1.689371735720014, −0.7457324405588255, 0,
0.7457324405588255, 1.689371735720014, 2.703487377077499, 3.256116455285572, 3.960591507626830, 4.362476984906328, 5.371934418287358, 6.069257871926116, 6.543475331438847, 7.448632514826294, 7.532405990079204, 8.195611089831604, 8.758423711071202, 9.469385109945099, 9.801596439125486, 10.37269288002303, 11.19862914536143, 11.54742031107410, 11.86920607716773, 12.59859264958217, 12.93114185531709, 13.73815253729166, 14.44570906848090, 14.69118508054562, 15.62345916144850
