| L(s) = 1 | − 11-s + 6·13-s − 7·19-s − 23-s − 5·25-s + 10·29-s + 10·31-s + 6·37-s − 5·41-s + 4·43-s − 47-s + 3·53-s − 3·59-s − 9·61-s − 4·67-s + 12·71-s + 4·73-s − 2·79-s + 2·83-s + 18·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 1.66·13-s − 1.60·19-s − 0.208·23-s − 25-s + 1.85·29-s + 1.79·31-s + 0.986·37-s − 0.780·41-s + 0.609·43-s − 0.145·47-s + 0.412·53-s − 0.390·59-s − 1.15·61-s − 0.488·67-s + 1.42·71-s + 0.468·73-s − 0.225·79-s + 0.219·83-s + 1.90·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.615701793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.615701793\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 9 T + p T^{2} \) | 1.61.j |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.91983475751174, −13.45599515789090, −13.12938128908025, −12.50359085123237, −11.91765137904245, −11.59855903735060, −10.84667919255439, −10.54053614169373, −10.12019181477833, −9.453018311779362, −8.808744403040077, −8.310388512285302, −8.127745599505813, −7.423297777581056, −6.499085265240390, −6.253230959468373, −6.001642235152391, −4.971202582072243, −4.530250670366561, −3.971844293431557, −3.374363466915301, −2.635791834558364, −2.071017150098134, −1.220322085655292, −0.5683045844709219,
0.5683045844709219, 1.220322085655292, 2.071017150098134, 2.635791834558364, 3.374363466915301, 3.971844293431557, 4.530250670366561, 4.971202582072243, 6.001642235152391, 6.253230959468373, 6.499085265240390, 7.423297777581056, 8.127745599505813, 8.310388512285302, 8.808744403040077, 9.453018311779362, 10.12019181477833, 10.54053614169373, 10.84667919255439, 11.59855903735060, 11.91765137904245, 12.50359085123237, 13.12938128908025, 13.45599515789090, 13.91983475751174
