| L(s) = 1 | − 2·3-s − 5-s + 9-s + 11-s + 2·15-s + 4·17-s + 4·19-s + 6·23-s + 25-s + 4·27-s + 2·29-s − 2·33-s − 6·37-s + 10·41-s + 4·43-s − 45-s − 10·47-s − 8·51-s + 2·53-s − 55-s − 8·57-s + 4·59-s + 14·61-s + 2·67-s − 12·69-s + 4·71-s + 4·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.516·15-s + 0.970·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s − 0.348·33-s − 0.986·37-s + 1.56·41-s + 0.609·43-s − 0.149·45-s − 1.45·47-s − 1.12·51-s + 0.274·53-s − 0.134·55-s − 1.05·57-s + 0.520·59-s + 1.79·61-s + 0.244·67-s − 1.44·69-s + 0.474·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.295140451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295140451\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−16.51727268569085, −16.09818926792441, −15.60577296299757, −14.76260062283090, −14.33934411945865, −13.72152398722215, −12.74939488302827, −12.50322806623458, −11.82066498715776, −11.24734805030841, −11.04461708602105, −10.09768113031334, −9.698908329424315, −8.799722946785967, −8.245184420506002, −7.351626572234970, −6.957143816239601, −6.191003103500198, −5.443622346156657, −5.113586976499486, −4.248523428684121, −3.418963656472665, −2.702495928017239, −1.302111686133630, −0.6432707623956495,
0.6432707623956495, 1.302111686133630, 2.702495928017239, 3.418963656472665, 4.248523428684121, 5.113586976499486, 5.443622346156657, 6.191003103500198, 6.957143816239601, 7.351626572234970, 8.245184420506002, 8.799722946785967, 9.698908329424315, 10.09768113031334, 11.04461708602105, 11.24734805030841, 11.82066498715776, 12.50322806623458, 12.74939488302827, 13.72152398722215, 14.33934411945865, 14.76260062283090, 15.60577296299757, 16.09818926792441, 16.51727268569085
