| L(s) = 1 | − 3-s + 3·5-s + 9-s − 11-s − 3·15-s + 2·19-s − 23-s + 4·25-s − 27-s − 6·29-s + 3·31-s + 33-s + 10·37-s + 12·41-s + 12·43-s + 3·45-s − 47-s − 7·49-s + 6·53-s − 3·55-s − 2·57-s + 8·59-s − 6·61-s + 9·67-s + 69-s − 71-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.301·11-s − 0.774·15-s + 0.458·19-s − 0.208·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.538·31-s + 0.174·33-s + 1.64·37-s + 1.87·41-s + 1.82·43-s + 0.447·45-s − 0.145·47-s − 49-s + 0.824·53-s − 0.404·55-s − 0.264·57-s + 1.04·59-s − 0.768·61-s + 1.09·67-s + 0.120·69-s − 0.118·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.296917677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.296917677\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.34137533831402, −12.80544633980404, −12.54341409442401, −11.91328289054438, −11.24994608510448, −10.97197951115106, −10.53736300610787, −9.754007742541881, −9.636598025975328, −9.260933298721276, −8.552089828861894, −7.781760820548381, −7.585052745815979, −6.808648701787271, −6.312327613176550, −5.820172267010259, −5.565489582769742, −5.006364286900128, −4.301814825934565, −3.881521563610539, −2.927888304016778, −2.426093770909830, −1.945469904289093, −1.112502158177952, −0.6129417116620687,
0.6129417116620687, 1.112502158177952, 1.945469904289093, 2.426093770909830, 2.927888304016778, 3.881521563610539, 4.301814825934565, 5.006364286900128, 5.565489582769742, 5.820172267010259, 6.312327613176550, 6.808648701787271, 7.585052745815979, 7.781760820548381, 8.552089828861894, 9.260933298721276, 9.636598025975328, 9.754007742541881, 10.53736300610787, 10.97197951115106, 11.24994608510448, 11.91328289054438, 12.54341409442401, 12.80544633980404, 13.34137533831402
