| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s − 4·11-s + 13-s + 2·15-s + 17-s + 8·19-s + 2·21-s − 25-s + 27-s + 2·29-s + 6·31-s − 4·33-s + 4·35-s − 4·37-s + 39-s + 2·41-s + 4·43-s + 2·45-s + 6·47-s − 3·49-s + 51-s + 6·53-s − 8·55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.516·15-s + 0.242·17-s + 1.83·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.696·33-s + 0.676·35-s − 0.657·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 0.875·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s − 1.07·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.790556263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.790556263\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−16.40381101611821, −15.92014314065443, −15.45557732106174, −14.78180542983223, −14.04798366849391, −13.78293927628797, −13.36406137028003, −12.62700604128514, −11.90253748874008, −11.38807250143418, −10.43698657509717, −10.21452648034515, −9.502158708224278, −8.899948293572046, −8.180576040074919, −7.668580281882211, −7.147709469789712, −6.123509425227531, −5.489247980650012, −5.025130475130475, −4.182421826163749, −3.151440673664317, −2.603930251980760, −1.767281500582274, −0.9278222189050818,
0.9278222189050818, 1.767281500582274, 2.603930251980760, 3.151440673664317, 4.182421826163749, 5.025130475130475, 5.489247980650012, 6.123509425227531, 7.147709469789712, 7.668580281882211, 8.180576040074919, 8.899948293572046, 9.502158708224278, 10.21452648034515, 10.43698657509717, 11.38807250143418, 11.90253748874008, 12.62700604128514, 13.36406137028003, 13.78293927628797, 14.04798366849391, 14.78180542983223, 15.45557732106174, 15.92014314065443, 16.40381101611821
