| L(s) = 1 | − 3-s + 5-s + 9-s − 5·13-s − 15-s − 6·17-s − 19-s + 4·23-s − 4·25-s − 27-s − 29-s + 10·31-s + 37-s + 5·39-s + 8·43-s + 45-s − 47-s + 6·51-s + 8·53-s + 57-s + 3·59-s − 6·61-s − 5·65-s + 13·67-s − 4·69-s − 3·73-s + 4·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.38·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s + 0.834·23-s − 4/5·25-s − 0.192·27-s − 0.185·29-s + 1.79·31-s + 0.164·37-s + 0.800·39-s + 1.21·43-s + 0.149·45-s − 0.145·47-s + 0.840·51-s + 1.09·53-s + 0.132·57-s + 0.390·59-s − 0.768·61-s − 0.620·65-s + 1.58·67-s − 0.481·69-s − 0.351·73-s + 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.532948975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.532948975\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.29289550614800, −13.01649084267851, −12.30631316179718, −12.05078194318934, −11.44618273214394, −11.05742583559552, −10.49172124032086, −10.04909544664467, −9.605833720666552, −9.120405290039940, −8.631802230490296, −7.957575902079256, −7.413607256946748, −6.976941511743255, −6.362901499815469, −6.114656818484056, −5.292223470199790, −4.936194173299937, −4.419046616716331, −3.928842405380027, −3.019930484415744, −2.266509214861757, −2.181328272709898, −1.071885343052907, −0.4185628910821925,
0.4185628910821925, 1.071885343052907, 2.181328272709898, 2.266509214861757, 3.019930484415744, 3.928842405380027, 4.419046616716331, 4.936194173299937, 5.292223470199790, 6.114656818484056, 6.362901499815469, 6.976941511743255, 7.413607256946748, 7.957575902079256, 8.631802230490296, 9.120405290039940, 9.605833720666552, 10.04909544664467, 10.49172124032086, 11.05742583559552, 11.44618273214394, 12.05078194318934, 12.30631316179718, 13.01649084267851, 13.29289550614800
