| L(s) = 1 | + 3-s − 7-s + 9-s + 4·11-s + 2·13-s + 17-s − 2·19-s − 21-s − 6·23-s + 27-s − 2·29-s + 10·31-s + 4·33-s − 8·37-s + 2·39-s − 8·43-s + 4·47-s + 49-s + 51-s − 6·53-s − 2·57-s − 4·59-s + 2·61-s − 63-s − 8·67-s − 6·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.242·17-s − 0.458·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s − 0.371·29-s + 1.79·31-s + 0.696·33-s − 1.31·37-s + 0.320·39-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.140·51-s − 0.824·53-s − 0.264·57-s − 0.520·59-s + 0.256·61-s − 0.125·63-s − 0.977·67-s − 0.722·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.13677337455351, −14.62395411253348, −14.08543652166045, −13.67107660806570, −13.28645937558167, −12.47789825844168, −12.02488705694733, −11.71214290263878, −10.86159694232564, −10.34178058726155, −9.803098557698348, −9.305824045792408, −8.747930796023905, −8.216801733764583, −7.789642845064341, −6.831474645154210, −6.562277024145577, −5.994760516401375, −5.221338733797694, −4.359671485818401, −3.919237284555140, −3.353304034910643, −2.636498971943698, −1.763905108237942, −1.202023085123048, 0,
1.202023085123048, 1.763905108237942, 2.636498971943698, 3.353304034910643, 3.919237284555140, 4.359671485818401, 5.221338733797694, 5.994760516401375, 6.562277024145577, 6.831474645154210, 7.789642845064341, 8.216801733764583, 8.747930796023905, 9.305824045792408, 9.803098557698348, 10.34178058726155, 10.86159694232564, 11.71214290263878, 12.02488705694733, 12.47789825844168, 13.28645937558167, 13.67107660806570, 14.08543652166045, 14.62395411253348, 15.13677337455351
