| L(s) = 1 | + 3-s − 2·5-s − 2·7-s + 9-s + 4·13-s − 2·15-s − 2·17-s + 2·19-s − 2·21-s − 25-s + 27-s − 10·29-s + 4·35-s + 10·37-s + 4·39-s − 6·41-s − 10·43-s − 2·45-s + 4·47-s − 3·49-s − 2·51-s + 2·53-s + 2·57-s − 8·59-s − 8·61-s − 2·63-s − 8·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.516·15-s − 0.485·17-s + 0.458·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.676·35-s + 1.64·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s − 0.298·45-s + 0.583·47-s − 3/7·49-s − 0.280·51-s + 0.274·53-s + 0.264·57-s − 1.04·59-s − 1.02·61-s − 0.251·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.217812983054236733103842082272, −7.84180560148000021637409394095, −6.91814068834537471384440133884, −6.25676705161894478498705998253, −5.28044023878169379311425856887, −4.08714030288010143683563873426, −3.65803310644861200687886466372, −2.81701283623621825978271365882, −1.52574618009013208365500770607, 0,
1.52574618009013208365500770607, 2.81701283623621825978271365882, 3.65803310644861200687886466372, 4.08714030288010143683563873426, 5.28044023878169379311425856887, 6.25676705161894478498705998253, 6.91814068834537471384440133884, 7.84180560148000021637409394095, 8.217812983054236733103842082272
