| L(s) = 1 | + 2·3-s + 9-s + 11-s + 2·13-s − 6·17-s + 2·19-s − 6·23-s − 4·27-s + 8·31-s + 2·33-s + 4·37-s + 4·39-s − 12·41-s − 4·43-s − 12·47-s − 12·51-s + 4·57-s − 2·61-s + 8·67-s − 12·69-s − 12·71-s + 2·73-s − 14·79-s − 11·81-s − 12·83-s − 6·89-s + 16·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s − 1.25·23-s − 0.769·27-s + 1.43·31-s + 0.348·33-s + 0.657·37-s + 0.640·39-s − 1.87·41-s − 0.609·43-s − 1.75·47-s − 1.68·51-s + 0.529·57-s − 0.256·61-s + 0.977·67-s − 1.44·69-s − 1.42·71-s + 0.234·73-s − 1.57·79-s − 1.22·81-s − 1.31·83-s − 0.635·89-s + 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.284576303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.284576303\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.09318319271983, −12.79319918351607, −11.84441380323384, −11.60983126850380, −11.34478901027772, −10.52302346524287, −10.00022912780551, −9.759133159027958, −9.133764139565962, −8.595182306455428, −8.359958630547078, −8.031281859822501, −7.281009458334523, −6.852319153353786, −6.222547874678279, −5.972165763213458, −5.084953049875918, −4.560081280578665, −4.096761298586483, −3.484855685749981, −3.056104354972018, −2.488930889231368, −1.834173605975854, −1.452563236194468, −0.3634785447444082,
0.3634785447444082, 1.452563236194468, 1.834173605975854, 2.488930889231368, 3.056104354972018, 3.484855685749981, 4.096761298586483, 4.560081280578665, 5.084953049875918, 5.972165763213458, 6.222547874678279, 6.852319153353786, 7.281009458334523, 8.031281859822501, 8.359958630547078, 8.595182306455428, 9.133764139565962, 9.759133159027958, 10.00022912780551, 10.52302346524287, 11.34478901027772, 11.60983126850380, 11.84441380323384, 12.79319918351607, 13.09318319271983
