| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 4·11-s − 6·13-s − 15-s + 4·17-s − 19-s − 2·21-s + 25-s − 27-s + 10·29-s − 2·31-s − 4·33-s + 2·35-s + 2·37-s + 6·39-s + 8·41-s + 8·43-s + 45-s − 3·49-s − 4·51-s + 6·53-s + 4·55-s + 57-s + 2·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.258·15-s + 0.970·17-s − 0.229·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.359·31-s − 0.696·33-s + 0.338·35-s + 0.328·37-s + 0.960·39-s + 1.24·41-s + 1.21·43-s + 0.149·45-s − 3/7·49-s − 0.560·51-s + 0.824·53-s + 0.539·55-s + 0.132·57-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.458481260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.458481260\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.96054643199627, −15.01048193183571, −14.61963034998665, −14.28360157886733, −13.76982089339347, −12.84212031728969, −12.32584988566911, −11.96951348102010, −11.48526385094659, −10.70225749731170, −10.25808023378222, −9.581588149791701, −9.221841732695502, −8.368419274971424, −7.723674796938047, −7.133122076024694, −6.572202729449075, −5.820867530558856, −5.337341979241082, −4.508449729725360, −4.258460317793206, −3.058587994737856, −2.334921134362806, −1.451861129528985, −0.7342964636926270,
0.7342964636926270, 1.451861129528985, 2.334921134362806, 3.058587994737856, 4.258460317793206, 4.508449729725360, 5.337341979241082, 5.820867530558856, 6.572202729449075, 7.133122076024694, 7.723674796938047, 8.368419274971424, 9.221841732695502, 9.581588149791701, 10.25808023378222, 10.70225749731170, 11.48526385094659, 11.96951348102010, 12.32584988566911, 12.84212031728969, 13.76982089339347, 14.28360157886733, 14.61963034998665, 15.01048193183571, 15.96054643199627
