| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s + 6·11-s − 14-s − 16-s − 2·17-s + 6·22-s + 28-s + 2·29-s − 8·31-s + 5·32-s − 2·34-s + 2·37-s − 4·41-s + 4·43-s − 6·44-s + 12·47-s + 49-s + 3·56-s + 2·58-s + 4·59-s − 14·61-s − 8·62-s + 7·64-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 1.80·11-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 1.27·22-s + 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s + 0.328·37-s − 0.624·41-s + 0.609·43-s − 0.904·44-s + 1.75·47-s + 1/7·49-s + 0.400·56-s + 0.262·58-s + 0.520·59-s − 1.79·61-s − 1.01·62-s + 7/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−13.14426794718797, −12.44294495996020, −12.16098049461135, −11.97512787102036, −11.14750312112172, −10.95001122688731, −10.18957404989268, −9.701921566985259, −9.173698976757465, −8.937997311600507, −8.665727810519230, −7.785225894290878, −7.336218792047464, −6.708018228562738, −6.326294970953983, −5.921523307724670, −5.380838346983880, −4.784456818753317, −4.217236910568485, −3.916490766941329, −3.460913949349963, −2.857052232561470, −2.173957887020701, −1.433293330327803, −0.7921438649146732, 0,
0.7921438649146732, 1.433293330327803, 2.173957887020701, 2.857052232561470, 3.460913949349963, 3.916490766941329, 4.217236910568485, 4.784456818753317, 5.380838346983880, 5.921523307724670, 6.326294970953983, 6.708018228562738, 7.336218792047464, 7.785225894290878, 8.665727810519230, 8.937997311600507, 9.173698976757465, 9.701921566985259, 10.18957404989268, 10.95001122688731, 11.14750312112172, 11.97512787102036, 12.16098049461135, 12.44294495996020, 13.14426794718797
