| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 13-s + 15-s − 6·17-s + 4·19-s + 21-s + 25-s − 27-s + 6·29-s − 8·31-s + 35-s + 2·37-s − 39-s + 6·41-s − 8·43-s − 45-s − 12·47-s + 49-s + 6·51-s − 6·53-s − 4·57-s + 12·59-s + 14·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.169·35-s + 0.328·37-s − 0.160·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s + 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−15.94426671223946, −15.51317399471442, −14.64946930441484, −14.35843069573847, −13.44864389066042, −13.02275010048079, −12.69898212634351, −11.72129228159837, −11.59469589684158, −10.94487590389582, −10.44892019917694, −9.640972233340439, −9.331378536149141, −8.440849669563933, −8.075254558672935, −7.169047144372385, −6.725958870970600, −6.286187588739806, −5.348698796367694, −4.970578496805958, −4.118375262993654, −3.602286432379060, −2.766403760211819, −1.903258212697719, −0.9023217311184532, 0,
0.9023217311184532, 1.903258212697719, 2.766403760211819, 3.602286432379060, 4.118375262993654, 4.970578496805958, 5.348698796367694, 6.286187588739806, 6.725958870970600, 7.169047144372385, 8.075254558672935, 8.440849669563933, 9.331378536149141, 9.640972233340439, 10.44892019917694, 10.94487590389582, 11.59469589684158, 11.72129228159837, 12.69898212634351, 13.02275010048079, 13.44864389066042, 14.35843069573847, 14.64946930441484, 15.51317399471442, 15.94426671223946
