| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s − 6·11-s − 2·15-s − 6·17-s + 2·19-s − 8·21-s − 6·23-s + 25-s − 4·27-s − 6·29-s + 2·31-s − 12·33-s + 4·35-s − 2·37-s + 6·41-s − 2·43-s − 45-s − 12·47-s + 9·49-s − 12·51-s + 6·53-s + 6·55-s + 4·57-s + 6·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.80·11-s − 0.516·15-s − 1.45·17-s + 0.458·19-s − 1.74·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.359·31-s − 2.08·33-s + 0.676·35-s − 0.328·37-s + 0.937·41-s − 0.304·43-s − 0.149·45-s − 1.75·47-s + 9/7·49-s − 1.68·51-s + 0.824·53-s + 0.809·55-s + 0.529·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5547104566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5547104566\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.97796826397965, −15.75657500637616, −15.11246101747413, −14.65596169738442, −13.70467893330868, −13.41698099358013, −13.02760374536561, −12.50118412864740, −11.66457960435713, −11.01640435078791, −10.31613438318567, −9.781675953307410, −9.299256768372595, −8.596407405373682, −8.089124401671059, −7.537655389370011, −6.918338291258120, −6.145404790211801, −5.488804064235576, −4.594502268440532, −3.740479446045345, −3.273007888932460, −2.540152697631834, −2.094808624907436, −0.2808779277031546,
0.2808779277031546, 2.094808624907436, 2.540152697631834, 3.273007888932460, 3.740479446045345, 4.594502268440532, 5.488804064235576, 6.145404790211801, 6.918338291258120, 7.537655389370011, 8.089124401671059, 8.596407405373682, 9.299256768372595, 9.781675953307410, 10.31613438318567, 11.01640435078791, 11.66457960435713, 12.50118412864740, 13.02760374536561, 13.41698099358013, 13.70467893330868, 14.65596169738442, 15.11246101747413, 15.75657500637616, 15.97796826397965
