| L(s) = 1 | − 3-s − 4·7-s + 9-s − 13-s + 2·17-s − 4·19-s + 4·21-s + 2·23-s − 27-s − 6·29-s − 2·31-s + 2·37-s + 39-s + 6·41-s + 12·43-s − 8·47-s + 9·49-s − 2·51-s + 6·53-s + 4·57-s + 4·59-s − 10·61-s − 4·63-s − 14·67-s − 2·69-s + 6·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.485·17-s − 0.917·19-s + 0.872·21-s + 0.417·23-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.328·37-s + 0.160·39-s + 0.937·41-s + 1.82·43-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s − 1.28·61-s − 0.503·63-s − 1.71·67-s − 0.240·69-s + 0.712·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7118637020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7118637020\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.32134268229895, −12.84549659742910, −12.70194786519476, −12.24598074746690, −11.53614539036354, −11.14568590719578, −10.56296410491148, −10.16827160547051, −9.681208235034929, −9.105304139703019, −8.938018065384209, −7.976105377076302, −7.467559542549199, −7.073366715796840, −6.445334277421540, −6.007355268634207, −5.682205585663469, −4.974326993649821, −4.275857972966937, −3.846578513594677, −3.188767089271237, −2.638615953057737, −1.956010549235294, −1.050251176203937, −0.2983229555172483,
0.2983229555172483, 1.050251176203937, 1.956010549235294, 2.638615953057737, 3.188767089271237, 3.846578513594677, 4.275857972966937, 4.974326993649821, 5.682205585663469, 6.007355268634207, 6.445334277421540, 7.073366715796840, 7.467559542549199, 7.976105377076302, 8.938018065384209, 9.105304139703019, 9.681208235034929, 10.16827160547051, 10.56296410491148, 11.14568590719578, 11.53614539036354, 12.24598074746690, 12.70194786519476, 12.84549659742910, 13.32134268229895
