| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 4·11-s + 5·13-s + 15-s − 8·17-s − 5·19-s − 21-s − 8·23-s + 25-s − 27-s + 3·29-s + 31-s + 4·33-s − 35-s − 8·37-s − 5·39-s − 11·41-s − 43-s − 45-s − 10·47-s − 6·49-s + 8·51-s + 2·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.38·13-s + 0.258·15-s − 1.94·17-s − 1.14·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 0.179·31-s + 0.696·33-s − 0.169·35-s − 1.31·37-s − 0.800·39-s − 1.71·41-s − 0.152·43-s − 0.149·45-s − 1.45·47-s − 6/7·49-s + 1.12·51-s + 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 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
−15.35500030194785, −15.00404201051045, −14.06554924915108, −13.64713958451997, −13.14246401753476, −12.78121244620579, −12.01890703428888, −11.61390723414502, −10.94761454588577, −10.76062937385589, −10.22694930311868, −9.562521676606333, −8.576176908765496, −8.363185701758371, −8.067737531601678, −7.061725582204580, −6.507094180341809, −6.265280863452112, −5.315420171604750, −4.923315731452277, −4.171858431322107, −3.811998707327230, −2.854253085019860, −2.029287301483190, −1.505414301187019, 0, 0,
1.505414301187019, 2.029287301483190, 2.854253085019860, 3.811998707327230, 4.171858431322107, 4.923315731452277, 5.315420171604750, 6.265280863452112, 6.507094180341809, 7.061725582204580, 8.067737531601678, 8.363185701758371, 8.576176908765496, 9.562521676606333, 10.22694930311868, 10.76062937385589, 10.94761454588577, 11.61390723414502, 12.01890703428888, 12.78121244620579, 13.14246401753476, 13.64713958451997, 14.06554924915108, 15.00404201051045, 15.35500030194785
