| L(s) = 1 | − 4·13-s + 2·17-s − 4·29-s + 12·37-s − 8·41-s − 7·49-s + 14·53-s + 10·61-s − 16·73-s + 16·89-s + 8·97-s + 20·101-s − 6·109-s + 14·113-s + ⋯ |
| L(s) = 1 | − 1.10·13-s + 0.485·17-s − 0.742·29-s + 1.97·37-s − 1.24·41-s − 49-s + 1.92·53-s + 1.28·61-s − 1.87·73-s + 1.69·89-s + 0.812·97-s + 1.99·101-s − 0.574·109-s + 1.31·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.685308444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.685308444\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−7.77033659390769825694978523224, −7.35571306364815264622875456391, −6.56045895951290303607717160923, −5.78182041923250559090991217656, −5.10341414108614073634161448242, −4.39951239405077178153987196967, −3.54035684878757153089658591985, −2.68638467477891280912996672684, −1.87657910627637354126634731065, −0.64662300259891474138093079546,
0.64662300259891474138093079546, 1.87657910627637354126634731065, 2.68638467477891280912996672684, 3.54035684878757153089658591985, 4.39951239405077178153987196967, 5.10341414108614073634161448242, 5.78182041923250559090991217656, 6.56045895951290303607717160923, 7.35571306364815264622875456391, 7.77033659390769825694978523224
