| L(s) = 1 | + 2·7-s − 5·13-s − 6·17-s − 6·19-s − 5·25-s − 9·29-s + 3·31-s + 8·37-s − 3·41-s + 8·43-s − 7·47-s − 3·49-s − 2·53-s − 4·59-s + 10·61-s − 8·67-s − 7·71-s + 9·73-s + 6·79-s − 14·83-s + 16·89-s − 10·91-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.38·13-s − 1.45·17-s − 1.37·19-s − 25-s − 1.67·29-s + 0.538·31-s + 1.31·37-s − 0.468·41-s + 1.21·43-s − 1.02·47-s − 3/7·49-s − 0.274·53-s − 0.520·59-s + 1.28·61-s − 0.977·67-s − 0.830·71-s + 1.05·73-s + 0.675·79-s − 1.53·83-s + 1.69·89-s − 1.04·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8441970051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8441970051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−14.85824015636086, −14.50799994192691, −13.80892450556438, −13.14149191841095, −12.91552579360697, −12.21011638174522, −11.62498349854613, −11.14172393514391, −10.81512920465902, −10.00941485972715, −9.545306887238372, −9.017330175884930, −8.364714287629903, −7.832223340638508, −7.384973655150225, −6.672852707694588, −6.152078552926645, −5.457287936826422, −4.719250310632100, −4.392233557905473, −3.770669521698627, −2.668281104918745, −2.183704004804005, −1.646273181218266, −0.3162800656969463,
0.3162800656969463, 1.646273181218266, 2.183704004804005, 2.668281104918745, 3.770669521698627, 4.392233557905473, 4.719250310632100, 5.457287936826422, 6.152078552926645, 6.672852707694588, 7.384973655150225, 7.832223340638508, 8.364714287629903, 9.017330175884930, 9.545306887238372, 10.00941485972715, 10.81512920465902, 11.14172393514391, 11.62498349854613, 12.21011638174522, 12.91552579360697, 13.14149191841095, 13.80892450556438, 14.50799994192691, 14.85824015636086
