| L(s) = 1 | + 3-s − 7-s + 9-s + 2·13-s − 6·17-s + 4·19-s − 21-s − 4·23-s + 27-s + 6·29-s + 8·31-s + 10·37-s + 2·39-s − 10·41-s + 12·43-s − 8·47-s + 49-s − 6·51-s − 6·53-s + 4·57-s − 4·59-s − 10·61-s − 63-s + 12·67-s − 4·69-s − 4·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.64·37-s + 0.320·39-s − 1.56·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s − 1.28·61-s − 0.125·63-s + 1.46·67-s − 0.481·69-s − 0.474·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.431165610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.431165610\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 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.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.935260227241260482688501882750, −7.11026965459911806140878402150, −6.37492595161886531828954868203, −5.97498275642496758586238868685, −4.72512551128399726914726315955, −4.35718953426952241212411641280, −3.34664745328940251179695592897, −2.75375086937936710337042905804, −1.85606743077579410936808758941, −0.74361469447525621027734926346,
0.74361469447525621027734926346, 1.85606743077579410936808758941, 2.75375086937936710337042905804, 3.34664745328940251179695592897, 4.35718953426952241212411641280, 4.72512551128399726914726315955, 5.97498275642496758586238868685, 6.37492595161886531828954868203, 7.11026965459911806140878402150, 7.935260227241260482688501882750
