| L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s − 2·13-s − 15-s + 6·17-s − 4·19-s − 4·23-s + 25-s − 27-s + 10·29-s + 4·31-s − 4·33-s + 2·37-s + 2·39-s − 2·41-s + 4·43-s + 45-s − 4·47-s − 7·49-s − 6·51-s − 10·53-s + 4·55-s + 4·57-s + 12·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s − 49-s − 0.840·51-s − 1.37·53-s + 0.539·55-s + 0.529·57-s + 1.56·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.460391158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.460391158\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 67 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.23414888411515, −13.88180679013515, −13.18202490077445, −12.52214619310744, −12.22839025847760, −11.80885042352533, −11.31759663891602, −10.53259654485388, −10.21944230087083, −9.658021855099341, −9.356854230293133, −8.447095743874210, −8.141670317040368, −7.444314479016081, −6.750127704470593, −6.235966683546338, −6.082792401318204, −5.147124690325777, −4.733982934880677, −4.126593118333089, −3.437360214347885, −2.748832960150725, −1.934828493678922, −1.275616700597487, −0.5927593818700439,
0.5927593818700439, 1.275616700597487, 1.934828493678922, 2.748832960150725, 3.437360214347885, 4.126593118333089, 4.733982934880677, 5.147124690325777, 6.082792401318204, 6.235966683546338, 6.750127704470593, 7.444314479016081, 8.141670317040368, 8.447095743874210, 9.356854230293133, 9.658021855099341, 10.21944230087083, 10.53259654485388, 11.31759663891602, 11.80885042352533, 12.22839025847760, 12.52214619310744, 13.18202490077445, 13.88180679013515, 14.23414888411515
