| L(s) = 1 | + 3-s − 3·7-s + 9-s + 3·11-s + 13-s − 3·17-s − 3·21-s − 4·23-s + 27-s − 5·29-s − 3·31-s + 3·33-s − 12·37-s + 39-s + 2·41-s + 4·43-s − 3·47-s + 2·49-s − 3·51-s + 9·53-s − 15·59-s + 3·61-s − 3·63-s − 7·67-s − 4·69-s − 8·71-s + 16·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.727·17-s − 0.654·21-s − 0.834·23-s + 0.192·27-s − 0.928·29-s − 0.538·31-s + 0.522·33-s − 1.97·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s − 0.437·47-s + 2/7·49-s − 0.420·51-s + 1.23·53-s − 1.95·59-s + 0.384·61-s − 0.377·63-s − 0.855·67-s − 0.481·69-s − 0.949·71-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.535926744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535926744\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−14.29074662688061, −13.65887206399039, −13.37993903686872, −12.83528864725377, −12.18288209045325, −12.00702102840249, −11.11825046344700, −10.70591645716436, −10.11512342732432, −9.493257153276684, −9.197122405644469, −8.741397271213435, −8.152567437842642, −7.397042262355220, −6.995073782097936, −6.414759329757661, −5.986907594432704, −5.301446672680015, −4.459193482407083, −3.847337462758241, −3.522222641398398, −2.850272729178865, −2.033576969822022, −1.518834280669907, −0.3901048179742146,
0.3901048179742146, 1.518834280669907, 2.033576969822022, 2.850272729178865, 3.522222641398398, 3.847337462758241, 4.459193482407083, 5.301446672680015, 5.986907594432704, 6.414759329757661, 6.995073782097936, 7.397042262355220, 8.152567437842642, 8.741397271213435, 9.197122405644469, 9.493257153276684, 10.11512342732432, 10.70591645716436, 11.11825046344700, 12.00702102840249, 12.18288209045325, 12.83528864725377, 13.37993903686872, 13.65887206399039, 14.29074662688061
