| L(s) = 1 | − 3-s − 7-s + 9-s + 3·11-s − 4·13-s − 6·17-s + 4·19-s + 21-s − 3·23-s − 27-s + 3·29-s + 10·31-s − 3·33-s − 7·37-s + 4·39-s + 43-s + 12·47-s + 49-s + 6·51-s + 6·53-s − 4·57-s − 12·59-s − 4·61-s − 63-s + 7·67-s + 3·69-s − 9·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.625·23-s − 0.192·27-s + 0.557·29-s + 1.79·31-s − 0.522·33-s − 1.15·37-s + 0.640·39-s + 0.152·43-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s − 1.56·59-s − 0.512·61-s − 0.125·63-s + 0.855·67-s + 0.361·69-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−7.23915256388055099462194300876, −6.75297664078658204068174382797, −6.16905741700059173386850236262, −5.39120123478087251324152617762, −4.55895716039817160799900570664, −4.12276261195502704367245350558, −3.02063187788417866022173453576, −2.23292901772985955406122408998, −1.12143930669857060958743154696, 0,
1.12143930669857060958743154696, 2.23292901772985955406122408998, 3.02063187788417866022173453576, 4.12276261195502704367245350558, 4.55895716039817160799900570664, 5.39120123478087251324152617762, 6.16905741700059173386850236262, 6.75297664078658204068174382797, 7.23915256388055099462194300876
