| L(s) = 1 | + 2·5-s − 3·9-s + 4·11-s + 6·13-s + 2·17-s − 4·19-s + 23-s − 25-s + 2·29-s − 4·31-s + 2·37-s + 6·41-s + 12·43-s − 6·45-s − 12·47-s + 10·53-s + 8·55-s + 2·61-s + 12·65-s + 12·67-s − 8·71-s + 14·73-s − 8·79-s + 9·81-s + 4·83-s + 4·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s + 1.20·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.328·37-s + 0.937·41-s + 1.82·43-s − 0.894·45-s − 1.75·47-s + 1.37·53-s + 1.07·55-s + 0.256·61-s + 1.48·65-s + 1.46·67-s − 0.949·71-s + 1.63·73-s − 0.900·79-s + 81-s + 0.439·83-s + 0.433·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.689634779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.689634779\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 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
−14.13462289281124, −13.70985889881625, −13.10929265472572, −12.75605762876298, −12.07140858585396, −11.49098205088848, −11.10949076793953, −10.71247442272541, −10.04089470029961, −9.405369647110056, −9.080038653695425, −8.556715637303515, −8.163530314320958, −7.410205358294195, −6.562298285417389, −6.320395237260760, −5.783434797069306, −5.457359103906985, −4.526101612703273, −3.846784965531129, −3.503384464331818, −2.626273459823039, −2.034127066825782, −1.305001155018095, −0.6788415626673604,
0.6788415626673604, 1.305001155018095, 2.034127066825782, 2.626273459823039, 3.503384464331818, 3.846784965531129, 4.526101612703273, 5.457359103906985, 5.783434797069306, 6.320395237260760, 6.562298285417389, 7.410205358294195, 8.163530314320958, 8.556715637303515, 9.080038653695425, 9.405369647110056, 10.04089470029961, 10.71247442272541, 11.10949076793953, 11.49098205088848, 12.07140858585396, 12.75605762876298, 13.10929265472572, 13.70985889881625, 14.13462289281124
