| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 2·13-s − 15-s + 2·17-s − 4·19-s + 4·21-s + 25-s − 27-s + 6·29-s + 8·31-s − 4·35-s + 6·37-s + 2·39-s + 10·41-s + 43-s + 45-s + 9·49-s − 2·51-s + 6·53-s + 4·57-s − 2·61-s − 4·63-s − 2·65-s − 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.676·35-s + 0.986·37-s + 0.320·39-s + 1.56·41-s + 0.152·43-s + 0.149·45-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.256·61-s − 0.503·63-s − 0.248·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.339709382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.339709382\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.72714516641768, −14.24561709857806, −13.53271269104652, −13.16446722624410, −12.63536951513616, −12.24685422022020, −11.75729299225476, −11.04081973513321, −10.30173708001235, −10.13628048676217, −9.634195287499582, −9.013137373682419, −8.480295766424484, −7.613464295887997, −7.153109610520778, −6.430258978292654, −6.091805786712343, −5.747126821195161, −4.689815116719277, −4.421637582591039, −3.517039177720757, −2.766590028716197, −2.394740616923033, −1.196298993012913, −0.4766272055870879,
0.4766272055870879, 1.196298993012913, 2.394740616923033, 2.766590028716197, 3.517039177720757, 4.421637582591039, 4.689815116719277, 5.747126821195161, 6.091805786712343, 6.430258978292654, 7.153109610520778, 7.613464295887997, 8.480295766424484, 9.013137373682419, 9.634195287499582, 10.13628048676217, 10.30173708001235, 11.04081973513321, 11.75729299225476, 12.24685422022020, 12.63536951513616, 13.16446722624410, 13.53271269104652, 14.24561709857806, 14.72714516641768
