| L(s) = 1 | + 3-s − 3·7-s − 2·9-s + 11-s + 2·13-s − 5·17-s + 19-s − 3·21-s − 2·23-s − 5·27-s − 3·29-s + 31-s + 33-s + 37-s + 2·39-s − 2·41-s − 4·43-s − 10·47-s + 2·49-s − 5·51-s + 13·53-s + 57-s + 14·59-s + 7·61-s + 6·63-s + 8·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s − 2/3·9-s + 0.301·11-s + 0.554·13-s − 1.21·17-s + 0.229·19-s − 0.654·21-s − 0.417·23-s − 0.962·27-s − 0.557·29-s + 0.179·31-s + 0.174·33-s + 0.164·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s − 1.45·47-s + 2/7·49-s − 0.700·51-s + 1.78·53-s + 0.132·57-s + 1.82·59-s + 0.896·61-s + 0.755·63-s + 0.977·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520644810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520644810\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−7.919961863132892380342078484993, −6.83028488045349002506276867624, −6.58470981007965405347898284348, −5.76737207613548130980850065002, −5.04631566766048524723150613294, −3.85929705177729365206322121501, −3.58901560474457578285263145828, −2.64822313695329361949103108782, −1.98228687757372789288788720936, −0.55214513931388672823899387776,
0.55214513931388672823899387776, 1.98228687757372789288788720936, 2.64822313695329361949103108782, 3.58901560474457578285263145828, 3.85929705177729365206322121501, 5.04631566766048524723150613294, 5.76737207613548130980850065002, 6.58470981007965405347898284348, 6.83028488045349002506276867624, 7.919961863132892380342078484993
