| L(s) = 1 | + 3·7-s + 3·11-s − 13-s + 2·17-s − 6·19-s − 7·23-s − 5·25-s − 9·29-s + 31-s − 5·37-s + 6·41-s − 7·43-s − 4·47-s + 2·49-s + 5·53-s − 3·59-s − 6·61-s − 12·67-s − 14·71-s − 6·73-s + 9·77-s − 14·79-s − 9·83-s + 13·89-s − 3·91-s + 2·97-s + 15·101-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.904·11-s − 0.277·13-s + 0.485·17-s − 1.37·19-s − 1.45·23-s − 25-s − 1.67·29-s + 0.179·31-s − 0.821·37-s + 0.937·41-s − 1.06·43-s − 0.583·47-s + 2/7·49-s + 0.686·53-s − 0.390·59-s − 0.768·61-s − 1.46·67-s − 1.66·71-s − 0.702·73-s + 1.02·77-s − 1.57·79-s − 0.987·83-s + 1.37·89-s − 0.314·91-s + 0.203·97-s + 1.49·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 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.ad |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.69229153628130630276751719546, −7.30821050768429622800522224567, −6.13003277986382822899782110235, −5.82200140244176234116239876050, −4.69523450338354479161851347756, −4.20028411602060003152898034325, −3.39188907659271894974302027392, −2.00818398982693796708435149461, −1.63341610205138870490692051440, 0,
1.63341610205138870490692051440, 2.00818398982693796708435149461, 3.39188907659271894974302027392, 4.20028411602060003152898034325, 4.69523450338354479161851347756, 5.82200140244176234116239876050, 6.13003277986382822899782110235, 7.30821050768429622800522224567, 7.69229153628130630276751719546
