| L(s) = 1 | − 3-s − 2·9-s + 13-s − 6·17-s + 2·19-s − 5·25-s + 5·27-s − 3·29-s − 5·31-s − 8·37-s − 39-s − 3·41-s − 8·43-s − 9·47-s + 6·51-s − 6·53-s − 2·57-s + 12·59-s + 14·61-s − 8·67-s − 15·71-s + 7·73-s + 5·75-s + 10·79-s + 81-s + 6·83-s + 3·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.277·13-s − 1.45·17-s + 0.458·19-s − 25-s + 0.962·27-s − 0.557·29-s − 0.898·31-s − 1.31·37-s − 0.160·39-s − 0.468·41-s − 1.21·43-s − 1.31·47-s + 0.840·51-s − 0.824·53-s − 0.264·57-s + 1.56·59-s + 1.79·61-s − 0.977·67-s − 1.78·71-s + 0.819·73-s + 0.577·75-s + 1.12·79-s + 1/9·81-s + 0.658·83-s + 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.23656066025336, −13.47810819528522, −13.38062464451379, −12.79616784273491, −12.07856047416734, −11.71351987882201, −11.28320681709904, −10.95740627072402, −10.33739907886347, −9.822543866150198, −9.268094320371294, −8.651237223386447, −8.412526353902126, −7.711742765997715, −7.045572756666276, −6.616882315224604, −6.153527214001959, −5.479551913763260, −5.132428712748030, −4.579907319679305, −3.661058833562488, −3.487659584329297, −2.505352484240448, −1.968260438114341, −1.261837868915541, 0, 0,
1.261837868915541, 1.968260438114341, 2.505352484240448, 3.487659584329297, 3.661058833562488, 4.579907319679305, 5.132428712748030, 5.479551913763260, 6.153527214001959, 6.616882315224604, 7.045572756666276, 7.711742765997715, 8.412526353902126, 8.651237223386447, 9.268094320371294, 9.822543866150198, 10.33739907886347, 10.95740627072402, 11.28320681709904, 11.71351987882201, 12.07856047416734, 12.79616784273491, 13.38062464451379, 13.47810819528522, 14.23656066025336
