| L(s) = 1 | − 3-s + 7-s + 9-s − 4·11-s + 6·13-s + 2·17-s − 8·19-s − 21-s − 4·23-s − 27-s + 6·29-s + 4·31-s + 4·33-s + 2·37-s − 6·39-s − 2·41-s + 12·43-s + 49-s − 2·51-s − 2·53-s + 8·57-s − 4·59-s + 6·61-s + 63-s + 4·67-s + 4·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.485·17-s − 1.83·19-s − 0.218·21-s − 0.834·23-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.696·33-s + 0.328·37-s − 0.960·39-s − 0.312·41-s + 1.82·43-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 1.05·57-s − 0.520·59-s + 0.768·61-s + 0.125·63-s + 0.488·67-s + 0.481·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473623629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473623629\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−8.283713073816522994800116171109, −7.87409351215571482688117694689, −6.79385122640234560145269351171, −6.08097445336548778967080520187, −5.62584976634956661875083466660, −4.57914663330558939890416727925, −4.04608270808290046286830210737, −2.88457105525048502422524416355, −1.88522155820502500331536252886, −0.71390979983163987945862546897,
0.71390979983163987945862546897, 1.88522155820502500331536252886, 2.88457105525048502422524416355, 4.04608270808290046286830210737, 4.57914663330558939890416727925, 5.62584976634956661875083466660, 6.08097445336548778967080520187, 6.79385122640234560145269351171, 7.87409351215571482688117694689, 8.283713073816522994800116171109
