| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 6·11-s + 15-s − 6·17-s + 4·19-s + 21-s + 25-s + 27-s − 6·29-s − 4·31-s + 6·33-s + 35-s + 4·37-s + 2·41-s − 4·43-s + 45-s − 2·47-s + 49-s − 6·51-s + 10·53-s + 6·55-s + 4·57-s + 14·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 1.04·33-s + 0.169·35-s + 0.657·37-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 0.291·47-s + 1/7·49-s − 0.840·51-s + 1.37·53-s + 0.809·55-s + 0.529·57-s + 1.82·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.391700140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.391700140\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.22632703972895, −13.63693296796621, −13.16272704534679, −12.86956528117784, −11.94168236150152, −11.66628856796299, −11.19335645352917, −10.64776759586149, −9.892863337940818, −9.371490228437310, −9.196403090002244, −8.561374645634024, −8.142474548188042, −7.252423567147020, −6.933516927870360, −6.499764911676765, −5.656278031921711, −5.314122931269972, −4.245207682869008, −4.159496467756178, −3.427810502198822, −2.652417567628748, −1.946840983575425, −1.507063004081404, −0.6787373495109072,
0.6787373495109072, 1.507063004081404, 1.946840983575425, 2.652417567628748, 3.427810502198822, 4.159496467756178, 4.245207682869008, 5.314122931269972, 5.656278031921711, 6.499764911676765, 6.933516927870360, 7.252423567147020, 8.142474548188042, 8.561374645634024, 9.196403090002244, 9.371490228437310, 9.892863337940818, 10.64776759586149, 11.19335645352917, 11.66628856796299, 11.94168236150152, 12.86956528117784, 13.16272704534679, 13.63693296796621, 14.22632703972895
