| L(s) = 1 | − 2·3-s − 4·7-s + 9-s + 6·11-s − 13-s + 6·17-s − 2·19-s + 8·21-s + 6·23-s + 4·27-s − 6·29-s − 2·31-s − 12·33-s − 2·37-s + 2·39-s − 6·41-s + 2·43-s − 12·47-s + 9·49-s − 12·51-s − 6·53-s + 4·57-s − 6·59-s + 2·61-s − 4·63-s − 4·67-s − 12·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s + 1.80·11-s − 0.277·13-s + 1.45·17-s − 0.458·19-s + 1.74·21-s + 1.25·23-s + 0.769·27-s − 1.11·29-s − 0.359·31-s − 2.08·33-s − 0.328·37-s + 0.320·39-s − 0.937·41-s + 0.304·43-s − 1.75·47-s + 9/7·49-s − 1.68·51-s − 0.824·53-s + 0.529·57-s − 0.781·59-s + 0.256·61-s − 0.503·63-s − 0.488·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8944437354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8944437354\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−8.213530218564666422042745170276, −7.06131301059558677772842905607, −6.71868250920883347037320613529, −6.09537021719183316071845549540, −5.49470951582837291168916858524, −4.63963070668118381856757068684, −3.56103682472085605806726816667, −3.18509658263545205045634750583, −1.60140924509725930425728921841, −0.56112236532926247126304648946,
0.56112236532926247126304648946, 1.60140924509725930425728921841, 3.18509658263545205045634750583, 3.56103682472085605806726816667, 4.63963070668118381856757068684, 5.49470951582837291168916858524, 6.09537021719183316071845549540, 6.71868250920883347037320613529, 7.06131301059558677772842905607, 8.213530218564666422042745170276
