| L(s) = 1 | + 3·3-s − 5-s + 5·7-s + 6·9-s + 3·11-s − 4·13-s − 3·15-s − 4·17-s + 4·19-s + 15·21-s + 4·23-s + 25-s + 9·27-s − 8·29-s + 2·31-s + 9·33-s − 5·35-s + 37-s − 12·39-s − 5·41-s + 10·43-s − 6·45-s − 7·47-s + 18·49-s − 12·51-s + 3·53-s − 3·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 1.88·7-s + 2·9-s + 0.904·11-s − 1.10·13-s − 0.774·15-s − 0.970·17-s + 0.917·19-s + 3.27·21-s + 0.834·23-s + 1/5·25-s + 1.73·27-s − 1.48·29-s + 0.359·31-s + 1.56·33-s − 0.845·35-s + 0.164·37-s − 1.92·39-s − 0.780·41-s + 1.52·43-s − 0.894·45-s − 1.02·47-s + 18/7·49-s − 1.68·51-s + 0.412·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.176909217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.176909217\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.632951475941198083078509945500, −8.119301489678652092988393658564, −7.31848503773726943554737082576, −7.09459373627450826055160195740, −5.42428671935800367224567580508, −4.53101965528081285358444824709, −4.07911853326119487339740564494, −2.99764691803540951519435754661, −2.11570282038458346113031184697, −1.34623803582289068316495601908,
1.34623803582289068316495601908, 2.11570282038458346113031184697, 2.99764691803540951519435754661, 4.07911853326119487339740564494, 4.53101965528081285358444824709, 5.42428671935800367224567580508, 7.09459373627450826055160195740, 7.31848503773726943554737082576, 8.119301489678652092988393658564, 8.632951475941198083078509945500
