| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 11-s + 15-s + 6·17-s + 6·19-s + 2·21-s + 5·23-s + 25-s − 27-s + 3·29-s + 7·31-s + 33-s + 2·35-s − 11·37-s − 10·41-s + 43-s − 45-s − 13·47-s − 3·49-s − 6·51-s + 10·53-s + 55-s − 6·57-s − 3·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 1.45·17-s + 1.37·19-s + 0.436·21-s + 1.04·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 1.25·31-s + 0.174·33-s + 0.338·35-s − 1.80·37-s − 1.56·41-s + 0.152·43-s − 0.149·45-s − 1.89·47-s − 3/7·49-s − 0.840·51-s + 1.37·53-s + 0.134·55-s − 0.794·57-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−15.23218000895818, −14.49304940090348, −13.94918920607617, −13.38911964577905, −12.97126106524780, −12.17367793893308, −12.01787304387595, −11.54814859777594, −10.81117748401943, −10.19222323030999, −9.919325773372750, −9.384534148090440, −8.511216767893115, −8.136940303861580, −7.405125629149773, −6.863471372490421, −6.535964913073721, −5.575577102576819, −5.249997080466575, −4.726175912778184, −3.752415381060191, −3.215647567370645, −2.849345246171071, −1.540458392401501, −0.9227578157397158, 0,
0.9227578157397158, 1.540458392401501, 2.849345246171071, 3.215647567370645, 3.752415381060191, 4.726175912778184, 5.249997080466575, 5.575577102576819, 6.535964913073721, 6.863471372490421, 7.405125629149773, 8.136940303861580, 8.511216767893115, 9.384534148090440, 9.919325773372750, 10.19222323030999, 10.81117748401943, 11.54814859777594, 12.01787304387595, 12.17367793893308, 12.97126106524780, 13.38911964577905, 13.94918920607617, 14.49304940090348, 15.23218000895818
