| L(s) = 1 | − 3-s + 7-s + 9-s − 4·11-s − 6·13-s + 2·17-s + 6·19-s − 21-s + 2·23-s − 27-s + 6·29-s − 2·31-s + 4·33-s − 4·37-s + 6·39-s + 8·41-s − 4·43-s + 4·47-s + 49-s − 2·51-s + 6·53-s − 6·57-s + 4·59-s + 14·61-s + 63-s + 4·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.485·17-s + 1.37·19-s − 0.218·21-s + 0.417·23-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.696·33-s − 0.657·37-s + 0.960·39-s + 1.24·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.794·57-s + 0.520·59-s + 1.79·61-s + 0.125·63-s + 0.488·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.239602017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239602017\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 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
−9.210404566701735127846868599033, −8.166336073132643556710806329904, −7.43457003082196541734923169077, −6.99199939874176212965959996776, −5.62684255626674712981984944920, −5.22049441389693591747869446709, −4.49676103610217940002555076348, −3.13827395936240739191356909048, −2.25153957288662252934621809731, −0.74506959536979126601875020491,
0.74506959536979126601875020491, 2.25153957288662252934621809731, 3.13827395936240739191356909048, 4.49676103610217940002555076348, 5.22049441389693591747869446709, 5.62684255626674712981984944920, 6.99199939874176212965959996776, 7.43457003082196541734923169077, 8.166336073132643556710806329904, 9.210404566701735127846868599033
