| L(s) = 1 | − 3-s − 2·9-s − 13-s + 6·17-s − 2·19-s + 23-s − 5·25-s + 5·27-s + 3·29-s + 5·31-s − 8·37-s + 39-s − 3·41-s + 8·43-s + 9·47-s − 6·51-s − 6·53-s + 2·57-s + 12·59-s + 14·61-s + 8·67-s − 69-s + 15·71-s + 7·73-s + 5·75-s + 10·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.277·13-s + 1.45·17-s − 0.458·19-s + 0.208·23-s − 25-s + 0.962·27-s + 0.557·29-s + 0.898·31-s − 1.31·37-s + 0.160·39-s − 0.468·41-s + 1.21·43-s + 1.31·47-s − 0.840·51-s − 0.824·53-s + 0.264·57-s + 1.56·59-s + 1.79·61-s + 0.977·67-s − 0.120·69-s + 1.78·71-s + 0.819·73-s + 0.577·75-s + 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.822931434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.822931434\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−14.05950987341732, −13.89709197761439, −12.89685742437545, −12.56523223062062, −12.06486805344605, −11.62040807676347, −11.23425712761602, −10.49876422184467, −10.17602380283016, −9.683644640366661, −9.001371169744819, −8.421601120609807, −8.023282010097904, −7.423194615243652, −6.762386166614807, −6.285103104685439, −5.689810324670366, −5.222807059368429, −4.824155528501040, −3.807721132761152, −3.556465028216352, −2.587612880978462, −2.183021522214314, −1.064467412824179, −0.5419579931591371,
0.5419579931591371, 1.064467412824179, 2.183021522214314, 2.587612880978462, 3.556465028216352, 3.807721132761152, 4.824155528501040, 5.222807059368429, 5.689810324670366, 6.285103104685439, 6.762386166614807, 7.423194615243652, 8.023282010097904, 8.421601120609807, 9.001371169744819, 9.683644640366661, 10.17602380283016, 10.49876422184467, 11.23425712761602, 11.62040807676347, 12.06486805344605, 12.56523223062062, 12.89685742437545, 13.89709197761439, 14.05950987341732
