| L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 2·13-s + 6·17-s − 4·19-s − 21-s − 27-s + 6·29-s + 4·31-s + 33-s + 2·37-s − 2·39-s − 6·41-s + 4·43-s + 49-s − 6·51-s − 6·53-s + 4·57-s + 12·59-s − 14·61-s + 63-s + 16·67-s − 2·73-s − 77-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s + 1.56·59-s − 1.79·61-s + 0.125·63-s + 1.95·67-s − 0.234·73-s − 0.113·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 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.o |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−12.71735130438739, −12.27002397651098, −11.77875248646068, −11.40280727404189, −10.97322832339718, −10.37913285079436, −10.14858541661742, −9.770973279974985, −8.997297415615314, −8.616640153187224, −8.086748209634002, −7.776813593322380, −7.224112208618144, −6.587533316400780, −6.235112854140062, −5.802041185917820, −5.229243616205677, −4.764196036915648, −4.399495849739169, −3.616382023847370, −3.317243930486031, −2.511766966312130, −2.028066930416400, −1.176565455126336, −0.9044928506736396, 0,
0.9044928506736396, 1.176565455126336, 2.028066930416400, 2.511766966312130, 3.317243930486031, 3.616382023847370, 4.399495849739169, 4.764196036915648, 5.229243616205677, 5.802041185917820, 6.235112854140062, 6.587533316400780, 7.224112208618144, 7.776813593322380, 8.086748209634002, 8.616640153187224, 8.997297415615314, 9.770973279974985, 10.14858541661742, 10.37913285079436, 10.97322832339718, 11.40280727404189, 11.77875248646068, 12.27002397651098, 12.71735130438739
