| L(s) = 1 | + 3·7-s − 3·11-s − 2·17-s + 2·19-s − 6·23-s − 5·25-s + 2·29-s + 4·31-s + 37-s − 7·41-s − 4·43-s + 47-s + 2·49-s − 9·53-s + 8·59-s − 4·61-s − 12·67-s − 5·71-s − 13·73-s − 9·77-s + 10·79-s − 83-s + 2·89-s − 12·97-s + 9·101-s + 8·103-s + 12·107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.904·11-s − 0.485·17-s + 0.458·19-s − 1.25·23-s − 25-s + 0.371·29-s + 0.718·31-s + 0.164·37-s − 1.09·41-s − 0.609·43-s + 0.145·47-s + 2/7·49-s − 1.23·53-s + 1.04·59-s − 0.512·61-s − 1.46·67-s − 0.593·71-s − 1.52·73-s − 1.02·77-s + 1.12·79-s − 0.109·83-s + 0.211·89-s − 1.21·97-s + 0.895·101-s + 0.788·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−7.999044503136549975821782066465, −7.28035725888149298372327579751, −6.32325445077640791758820442634, −5.60901788971499540980473984869, −4.85252998022211379195095218790, −4.29299538320593586189954154448, −3.23575962858943676662393916375, −2.25946589607229941605861686547, −1.49237900593426958795742047439, 0,
1.49237900593426958795742047439, 2.25946589607229941605861686547, 3.23575962858943676662393916375, 4.29299538320593586189954154448, 4.85252998022211379195095218790, 5.60901788971499540980473984869, 6.32325445077640791758820442634, 7.28035725888149298372327579751, 7.999044503136549975821782066465
