| L(s) = 1 | + 2-s − 4-s − 3·8-s − 4·13-s − 16-s − 4·17-s + 4·19-s + 8·23-s − 4·26-s + 2·29-s + 2·31-s + 5·32-s − 4·34-s − 6·37-s + 4·38-s − 6·41-s + 8·46-s + 12·47-s − 7·49-s + 4·52-s − 2·53-s + 2·58-s + 6·59-s + 14·61-s + 2·62-s + 7·64-s + 67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1.10·13-s − 1/4·16-s − 0.970·17-s + 0.917·19-s + 1.66·23-s − 0.784·26-s + 0.371·29-s + 0.359·31-s + 0.883·32-s − 0.685·34-s − 0.986·37-s + 0.648·38-s − 0.937·41-s + 1.17·46-s + 1.75·47-s − 49-s + 0.554·52-s − 0.274·53-s + 0.262·58-s + 0.781·59-s + 1.79·61-s + 0.254·62-s + 7/8·64-s + 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 67 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−16.14367833437343, −15.61179162558372, −15.14520194901889, −14.51690559101424, −14.17232359540222, −13.47258999645661, −13.04100785291285, −12.59553901310820, −11.77558394674380, −11.60945034651482, −10.66225219921205, −10.01115092992529, −9.516278150859875, −8.727296261582350, −8.568718161522289, −7.369246176992802, −7.067712870518372, −6.267909466828094, −5.457353664478093, −4.946794347590069, −4.538186550493084, −3.644108460962019, −2.985130612861974, −2.322655443963211, −1.060772533613975, 0,
1.060772533613975, 2.322655443963211, 2.985130612861974, 3.644108460962019, 4.538186550493084, 4.946794347590069, 5.457353664478093, 6.267909466828094, 7.067712870518372, 7.369246176992802, 8.568718161522289, 8.727296261582350, 9.516278150859875, 10.01115092992529, 10.66225219921205, 11.60945034651482, 11.77558394674380, 12.59553901310820, 13.04100785291285, 13.47258999645661, 14.17232359540222, 14.51690559101424, 15.14520194901889, 15.61179162558372, 16.14367833437343
