| L(s) = 1 | − 3·7-s − 2·13-s − 2·17-s − 3·19-s + 23-s − 5·25-s + 2·29-s + 5·31-s − 7·37-s − 4·43-s + 8·47-s + 2·49-s + 4·53-s + 6·59-s − 11·61-s − 5·67-s + 6·71-s + 5·73-s − 5·79-s + 4·83-s + 2·89-s + 6·91-s + 19·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.554·13-s − 0.485·17-s − 0.688·19-s + 0.208·23-s − 25-s + 0.371·29-s + 0.898·31-s − 1.15·37-s − 0.609·43-s + 1.16·47-s + 2/7·49-s + 0.549·53-s + 0.781·59-s − 1.40·61-s − 0.610·67-s + 0.712·71-s + 0.585·73-s − 0.562·79-s + 0.439·83-s + 0.211·89-s + 0.628·91-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−13.40377697906135, −12.78168269662388, −12.35436723675224, −12.00211284407368, −11.50736740271856, −10.88122786579639, −10.34357913493003, −10.06189925494890, −9.596000482343493, −8.983262285747534, −8.704284592614605, −8.073505888057303, −7.472501894459506, −7.056745406361341, −6.438730759208465, −6.226208326051121, −5.583937343142372, −4.929708363697114, −4.489254577568883, −3.784313105495362, −3.430119040939533, −2.660659556912018, −2.298308602560639, −1.555158445315228, −0.6198087323819761, 0,
0.6198087323819761, 1.555158445315228, 2.298308602560639, 2.660659556912018, 3.430119040939533, 3.784313105495362, 4.489254577568883, 4.929708363697114, 5.583937343142372, 6.226208326051121, 6.438730759208465, 7.056745406361341, 7.472501894459506, 8.073505888057303, 8.704284592614605, 8.983262285747534, 9.596000482343493, 10.06189925494890, 10.34357913493003, 10.88122786579639, 11.50736740271856, 12.00211284407368, 12.35436723675224, 12.78168269662388, 13.40377697906135
