| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 5·11-s + 5·13-s + 15-s − 3·17-s + 4·19-s + 21-s + 9·23-s + 25-s + 27-s − 3·29-s + 2·31-s − 5·33-s + 35-s + 10·37-s + 5·39-s − 10·41-s − 9·43-s + 45-s + 2·47-s − 6·49-s − 3·51-s − 14·53-s − 5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.38·13-s + 0.258·15-s − 0.727·17-s + 0.917·19-s + 0.218·21-s + 1.87·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s + 0.359·31-s − 0.870·33-s + 0.169·35-s + 1.64·37-s + 0.800·39-s − 1.56·41-s − 1.37·43-s + 0.149·45-s + 0.291·47-s − 6/7·49-s − 0.420·51-s − 1.92·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 79 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−14.28759282397725, −13.59677573056686, −13.30800750619530, −13.07793324644945, −12.58196932971257, −11.55383743130065, −11.31152784856593, −10.78692079661130, −10.35679271365515, −9.605807839650016, −9.362170155877124, −8.538338430522839, −8.325483615458595, −7.785560613242968, −7.115888131254852, −6.664377683735301, −5.939848396645167, −5.414882205583437, −4.804290678849243, −4.447579804211249, −3.285901009423963, −3.168015478126219, −2.453276138787025, −1.609487518653676, −1.162120403807625, 0,
1.162120403807625, 1.609487518653676, 2.453276138787025, 3.168015478126219, 3.285901009423963, 4.447579804211249, 4.804290678849243, 5.414882205583437, 5.939848396645167, 6.664377683735301, 7.115888131254852, 7.785560613242968, 8.325483615458595, 8.538338430522839, 9.362170155877124, 9.605807839650016, 10.35679271365515, 10.78692079661130, 11.31152784856593, 11.55383743130065, 12.58196932971257, 13.07793324644945, 13.30800750619530, 13.59677573056686, 14.28759282397725
