| L(s) = 1 | + 3-s + 2·5-s − 2·9-s + 5·11-s + 13-s + 2·15-s + 8·17-s − 8·19-s − 7·23-s − 25-s − 5·27-s + 6·29-s + 5·31-s + 5·33-s + 37-s + 39-s − 9·41-s − 2·43-s − 4·45-s − 3·47-s + 8·51-s + 10·53-s + 10·55-s − 8·57-s + 4·59-s + 7·61-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 2/3·9-s + 1.50·11-s + 0.277·13-s + 0.516·15-s + 1.94·17-s − 1.83·19-s − 1.45·23-s − 1/5·25-s − 0.962·27-s + 1.11·29-s + 0.898·31-s + 0.870·33-s + 0.164·37-s + 0.160·39-s − 1.40·41-s − 0.304·43-s − 0.596·45-s − 0.437·47-s + 1.12·51-s + 1.37·53-s + 1.34·55-s − 1.05·57-s + 0.520·59-s + 0.896·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.393016517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.393016517\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−16.73051103847570, −16.21844120937837, −15.19863461727388, −14.78504939643324, −14.23697225956256, −13.87672289838109, −13.43875862364278, −12.46480623590752, −12.00224871145127, −11.56606639734229, −10.57144387246484, −10.00425930212966, −9.629215115879555, −8.864293280632701, −8.243315319330288, −7.983862338465249, −6.632903372739053, −6.398102412620506, −5.735603895507623, −4.974707853570865, −3.833234992598847, −3.591111696879199, −2.413452318423422, −1.887776584923641, −0.8598748104182698,
0.8598748104182698, 1.887776584923641, 2.413452318423422, 3.591111696879199, 3.833234992598847, 4.974707853570865, 5.735603895507623, 6.398102412620506, 6.632903372739053, 7.983862338465249, 8.243315319330288, 8.864293280632701, 9.629215115879555, 10.00425930212966, 10.57144387246484, 11.56606639734229, 12.00224871145127, 12.46480623590752, 13.43875862364278, 13.87672289838109, 14.23697225956256, 14.78504939643324, 15.19863461727388, 16.21844120937837, 16.73051103847570
