| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s − 2·9-s − 10-s + 3·11-s − 12-s + 13-s + 15-s + 16-s − 2·18-s − 8·19-s − 20-s + 3·22-s + 7·23-s − 24-s + 25-s + 26-s + 5·27-s − 2·29-s + 30-s + 3·31-s + 32-s − 3·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.471·18-s − 1.83·19-s − 0.223·20-s + 0.639·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s − 0.371·29-s + 0.182·30-s + 0.538·31-s + 0.176·32-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.089190458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.089190458\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−8.058166232474312061011859548834, −6.81180626020961832494642353298, −6.74752147876504730264363458996, −5.85328222300210070519488775083, −5.17310558415713369929048092176, −4.42482099673736837190899133581, −3.74643680464159872039351646139, −2.94402730205184369088092330067, −1.92180014996785771525210010290, −0.68760031086648837926308031444,
0.68760031086648837926308031444, 1.92180014996785771525210010290, 2.94402730205184369088092330067, 3.74643680464159872039351646139, 4.42482099673736837190899133581, 5.17310558415713369929048092176, 5.85328222300210070519488775083, 6.74752147876504730264363458996, 6.81180626020961832494642353298, 8.058166232474312061011859548834
