| L(s) = 1 | + 5-s + 7-s − 11-s + 6·13-s + 2·17-s − 4·19-s + 25-s + 6·29-s + 4·31-s + 35-s − 2·37-s − 6·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s − 55-s − 2·61-s + 6·65-s + 12·67-s − 6·73-s − 77-s − 12·83-s + 2·85-s − 14·89-s + 6·91-s − 4·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.301·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.169·35-s − 0.328·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 0.256·61-s + 0.744·65-s + 1.46·67-s − 0.702·73-s − 0.113·77-s − 1.31·83-s + 0.216·85-s − 1.48·89-s + 0.628·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.803289021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.803289021\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.04340132120657, −12.53358007760310, −12.01303455077273, −11.57150077440784, −11.02750292152201, −10.64701586220440, −10.14546358452636, −9.905832606804448, −9.062287458373291, −8.572737363146673, −8.427486968429852, −7.913011163865394, −7.154673696021863, −6.689521806048266, −6.266288106138322, −5.672297139902972, −5.391760718440237, −4.564974550917287, −4.261390404890970, −3.529499403889615, −3.066238167418289, −2.376442597392812, −1.770027602596563, −1.182652348435572, −0.5800805835855275,
0.5800805835855275, 1.182652348435572, 1.770027602596563, 2.376442597392812, 3.066238167418289, 3.529499403889615, 4.261390404890970, 4.564974550917287, 5.391760718440237, 5.672297139902972, 6.266288106138322, 6.689521806048266, 7.154673696021863, 7.913011163865394, 8.427486968429852, 8.572737363146673, 9.062287458373291, 9.905832606804448, 10.14546358452636, 10.64701586220440, 11.02750292152201, 11.57150077440784, 12.01303455077273, 12.53358007760310, 13.04340132120657
