| L(s) = 1 | − 2·3-s + 9-s − 6·11-s − 6·17-s + 2·19-s + 4·27-s + 12·33-s − 6·41-s − 10·43-s + 12·51-s − 4·57-s + 6·59-s − 14·67-s − 2·73-s − 11·81-s − 18·83-s + 18·89-s + 10·97-s − 6·99-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.80·11-s − 1.45·17-s + 0.458·19-s + 0.769·27-s + 2.08·33-s − 0.937·41-s − 1.52·43-s + 1.68·51-s − 0.529·57-s + 0.781·59-s − 1.71·67-s − 0.234·73-s − 1.22·81-s − 1.97·83-s + 1.90·89-s + 1.01·97-s − 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−13.11359523530542, −12.64864136272972, −12.06016597582549, −11.70366485940006, −11.27487917548053, −10.86318690799248, −10.40855792836678, −10.14507959268102, −9.608062374464800, −8.789190211014373, −8.603881662485920, −7.981863355363021, −7.495861231448373, −6.921705665851070, −6.599350530660261, −5.950377443101717, −5.581498762192452, −5.066694375054230, −4.747065475336159, −4.280071664945133, −3.359890355636916, −2.957000028366813, −2.300591969724040, −1.775034583040299, −0.9034815345369263, 0, 0,
0.9034815345369263, 1.775034583040299, 2.300591969724040, 2.957000028366813, 3.359890355636916, 4.280071664945133, 4.747065475336159, 5.066694375054230, 5.581498762192452, 5.950377443101717, 6.599350530660261, 6.921705665851070, 7.495861231448373, 7.981863355363021, 8.603881662485920, 8.789190211014373, 9.608062374464800, 10.14507959268102, 10.40855792836678, 10.86318690799248, 11.27487917548053, 11.70366485940006, 12.06016597582549, 12.64864136272972, 13.11359523530542
