| L(s) = 1 | − 2·7-s + 11-s − 13-s − 3·17-s + 5·19-s − 6·23-s + 6·29-s + 2·31-s − 5·37-s − 9·41-s + 43-s − 3·47-s − 3·49-s + 12·59-s + 8·61-s − 5·67-s − 12·71-s − 2·73-s − 2·77-s − 10·79-s − 6·83-s + 2·91-s + 97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s − 0.277·13-s − 0.727·17-s + 1.14·19-s − 1.25·23-s + 1.11·29-s + 0.359·31-s − 0.821·37-s − 1.40·41-s + 0.152·43-s − 0.437·47-s − 3/7·49-s + 1.56·59-s + 1.02·61-s − 0.610·67-s − 1.42·71-s − 0.234·73-s − 0.227·77-s − 1.12·79-s − 0.658·83-s + 0.209·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138378449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138378449\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.41981115160370, −13.16068333028229, −12.45481387586926, −12.02228375623602, −11.65442550225181, −11.24549918104528, −10.34132467648276, −10.08768277665883, −9.774397916698371, −9.127798473935092, −8.495641794776128, −8.320116264416421, −7.411069373176721, −7.087484377430094, −6.511537156256726, −6.112587344138341, −5.464458541079983, −4.916431055468408, −4.333890769037835, −3.720857320004455, −3.185143907016318, −2.641873372338866, −1.915693981363827, −1.235473993527010, −0.3294208219150571,
0.3294208219150571, 1.235473993527010, 1.915693981363827, 2.641873372338866, 3.185143907016318, 3.720857320004455, 4.333890769037835, 4.916431055468408, 5.464458541079983, 6.112587344138341, 6.511537156256726, 7.087484377430094, 7.411069373176721, 8.320116264416421, 8.495641794776128, 9.127798473935092, 9.774397916698371, 10.08768277665883, 10.34132467648276, 11.24549918104528, 11.65442550225181, 12.02228375623602, 12.45481387586926, 13.16068333028229, 13.41981115160370
