| L(s) = 1 | + 3-s − 2·9-s + 11-s − 5·13-s − 6·17-s + 2·19-s − 6·23-s − 5·25-s − 5·27-s + 3·29-s + 8·31-s + 33-s + 2·37-s − 5·39-s + 6·41-s + 4·43-s + 6·47-s − 6·51-s − 12·53-s + 2·57-s − 3·59-s + 7·61-s + 13·67-s − 6·69-s + 12·71-s + 10·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.301·11-s − 1.38·13-s − 1.45·17-s + 0.458·19-s − 1.25·23-s − 25-s − 0.962·27-s + 0.557·29-s + 1.43·31-s + 0.174·33-s + 0.328·37-s − 0.800·39-s + 0.937·41-s + 0.609·43-s + 0.875·47-s − 0.840·51-s − 1.64·53-s + 0.264·57-s − 0.390·59-s + 0.896·61-s + 1.58·67-s − 0.722·69-s + 1.42·71-s + 1.17·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.649952975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.649952975\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 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 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−8.000025899604980278126983009867, −7.15250298312213630874458995237, −6.39296506503414023858608741118, −5.80483055563975946398701373276, −4.85148366100737723154744563169, −4.27015678142307257426467066148, −3.43298578700714820715582949343, −2.39310123480641844699802744043, −2.19902236699105380741450477916, −0.57139773279138586482545680334,
0.57139773279138586482545680334, 2.19902236699105380741450477916, 2.39310123480641844699802744043, 3.43298578700714820715582949343, 4.27015678142307257426467066148, 4.85148366100737723154744563169, 5.80483055563975946398701373276, 6.39296506503414023858608741118, 7.15250298312213630874458995237, 8.000025899604980278126983009867
