| L(s) = 1 | − 3-s − 2·4-s − 2·7-s + 9-s − 3·11-s + 2·12-s + 4·13-s + 4·16-s + 5·19-s + 2·21-s − 27-s + 4·28-s − 3·29-s + 8·31-s + 3·33-s − 2·36-s − 2·37-s − 4·39-s − 3·41-s + 10·43-s + 6·44-s + 6·47-s − 4·48-s − 3·49-s − 8·52-s + 6·53-s − 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.755·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 1.10·13-s + 16-s + 1.14·19-s + 0.436·21-s − 0.192·27-s + 0.755·28-s − 0.557·29-s + 1.43·31-s + 0.522·33-s − 1/3·36-s − 0.328·37-s − 0.640·39-s − 0.468·41-s + 1.52·43-s + 0.904·44-s + 0.875·47-s − 0.577·48-s − 3/7·49-s − 1.10·52-s + 0.824·53-s − 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.159349077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.159349077\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 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
−15.70714030841742, −15.12473231897527, −14.34757778760801, −13.72238761705308, −13.35750772589187, −12.98209705727052, −12.36818346698969, −11.76152338514420, −11.20292660014818, −10.35532470124443, −10.16136469725807, −9.525095812985585, −8.844638636898088, −8.422417560451620, −7.612922866864522, −7.154895045445736, −6.187862552386189, −5.812106891790217, −5.219971010675041, −4.580767096019389, −3.805989844471957, −3.311123726559453, −2.439854737422739, −1.153529788347155, −0.5435139573909733,
0.5435139573909733, 1.153529788347155, 2.439854737422739, 3.311123726559453, 3.805989844471957, 4.580767096019389, 5.219971010675041, 5.812106891790217, 6.187862552386189, 7.154895045445736, 7.612922866864522, 8.422417560451620, 8.844638636898088, 9.525095812985585, 10.16136469725807, 10.35532470124443, 11.20292660014818, 11.76152338514420, 12.36818346698969, 12.98209705727052, 13.35750772589187, 13.72238761705308, 14.34757778760801, 15.12473231897527, 15.70714030841742
