| L(s) = 1 | − 3-s + 2·7-s − 2·9-s + 4·11-s + 5·13-s + 2·17-s − 6·19-s − 2·21-s + 23-s + 5·27-s + 29-s + 9·31-s − 4·33-s + 4·37-s − 5·39-s + 3·41-s + 8·43-s − 5·47-s − 3·49-s − 2·51-s − 6·53-s + 6·57-s + 4·59-s − 10·61-s − 4·63-s − 4·67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s − 2/3·9-s + 1.20·11-s + 1.38·13-s + 0.485·17-s − 1.37·19-s − 0.436·21-s + 0.208·23-s + 0.962·27-s + 0.185·29-s + 1.61·31-s − 0.696·33-s + 0.657·37-s − 0.800·39-s + 0.468·41-s + 1.21·43-s − 0.729·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s + 0.794·57-s + 0.520·59-s − 1.28·61-s − 0.503·63-s − 0.488·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.122041291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.122041291\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.985658589521102488821191331844, −6.74540670132286989511050735167, −6.27690489847011655287345975999, −5.88386890791295540599767669414, −4.89461207318142856187001601883, −4.31324998208636068643982276976, −3.56643450939491712500973188331, −2.60494147301632000549227948048, −1.51305505368215408318377373579, −0.790168275053358464574806541640,
0.790168275053358464574806541640, 1.51305505368215408318377373579, 2.60494147301632000549227948048, 3.56643450939491712500973188331, 4.31324998208636068643982276976, 4.89461207318142856187001601883, 5.88386890791295540599767669414, 6.27690489847011655287345975999, 6.74540670132286989511050735167, 7.985658589521102488821191331844
