| L(s) = 1 | − 3-s + 2·7-s + 9-s + 4·13-s + 2·17-s − 19-s − 2·21-s + 6·23-s − 27-s − 2·29-s + 4·37-s − 4·39-s + 2·41-s − 2·43-s + 6·47-s − 3·49-s − 2·51-s + 12·53-s + 57-s − 4·59-s + 6·61-s + 2·63-s − 12·67-s − 6·69-s + 2·73-s + 81-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.10·13-s + 0.485·17-s − 0.229·19-s − 0.436·21-s + 1.25·23-s − 0.192·27-s − 0.371·29-s + 0.657·37-s − 0.640·39-s + 0.312·41-s − 0.304·43-s + 0.875·47-s − 3/7·49-s − 0.280·51-s + 1.64·53-s + 0.132·57-s − 0.520·59-s + 0.768·61-s + 0.251·63-s − 1.46·67-s − 0.722·69-s + 0.234·73-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.021923360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.021923360\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−8.121962773207804684801184662954, −7.38041849681696519921818884163, −6.69798136201269317955576882250, −5.86450497854006685205296629707, −5.34760397113613354355040360032, −4.51305297219797250543328799208, −3.81155307263829007348655142833, −2.81338268548254459630772850306, −1.64071218034734377049283055699, −0.840306604869407963964462634276,
0.840306604869407963964462634276, 1.64071218034734377049283055699, 2.81338268548254459630772850306, 3.81155307263829007348655142833, 4.51305297219797250543328799208, 5.34760397113613354355040360032, 5.86450497854006685205296629707, 6.69798136201269317955576882250, 7.38041849681696519921818884163, 8.121962773207804684801184662954
