| L(s) = 1 | + 2·5-s + 5·7-s + 11-s − 2·13-s − 7·17-s − 23-s − 25-s − 3·29-s + 8·31-s + 10·35-s − 3·37-s + 11·41-s + 9·43-s − 47-s + 18·49-s + 12·53-s + 2·55-s + 5·59-s + 6·61-s − 4·65-s + 4·67-s + 4·73-s + 5·77-s − 5·79-s − 6·83-s − 14·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.88·7-s + 0.301·11-s − 0.554·13-s − 1.69·17-s − 0.208·23-s − 1/5·25-s − 0.557·29-s + 1.43·31-s + 1.69·35-s − 0.493·37-s + 1.71·41-s + 1.37·43-s − 0.145·47-s + 18/7·49-s + 1.64·53-s + 0.269·55-s + 0.650·59-s + 0.768·61-s − 0.496·65-s + 0.488·67-s + 0.468·73-s + 0.569·77-s − 0.562·79-s − 0.658·83-s − 1.51·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.934889145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.934889145\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−8.361002662824019684742742815425, −7.58882230854122823807401845925, −6.90618001934064826340600378038, −5.98840650159256748048828140308, −5.36072865732906606518411738987, −4.55291776181200437307107737786, −4.09049126652147869868582350696, −2.36581218765512307123006075028, −2.13604412215569288894445188264, −0.990712056161014381484734803862,
0.990712056161014381484734803862, 2.13604412215569288894445188264, 2.36581218765512307123006075028, 4.09049126652147869868582350696, 4.55291776181200437307107737786, 5.36072865732906606518411738987, 5.98840650159256748048828140308, 6.90618001934064826340600378038, 7.58882230854122823807401845925, 8.361002662824019684742742815425
