| L(s) = 1 | − 5-s − 11-s − 6·13-s + 4·17-s − 6·19-s + 3·23-s − 4·25-s + 4·29-s + 9·31-s + 7·37-s + 2·41-s − 6·43-s + 12·47-s − 7·49-s − 2·53-s + 55-s + 9·59-s + 8·61-s + 6·65-s + 15·67-s − 3·71-s − 6·73-s + 6·79-s − 6·83-s − 4·85-s + 5·89-s + 6·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.66·13-s + 0.970·17-s − 1.37·19-s + 0.625·23-s − 4/5·25-s + 0.742·29-s + 1.61·31-s + 1.15·37-s + 0.312·41-s − 0.914·43-s + 1.75·47-s − 49-s − 0.274·53-s + 0.134·55-s + 1.17·59-s + 1.02·61-s + 0.744·65-s + 1.83·67-s − 0.356·71-s − 0.702·73-s + 0.675·79-s − 0.658·83-s − 0.433·85-s + 0.529·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321755596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321755596\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−8.429190100092190501442904145583, −8.018368082086072034023308252606, −7.22419907225686154066826773454, −6.52728041576470757773561459975, −5.56498681784481166892228678633, −4.75044742748289031222922601625, −4.10715158347972286015649299645, −2.93913571733158207691707003027, −2.22938384024183425897284539951, −0.67220471180945196290978612656,
0.67220471180945196290978612656, 2.22938384024183425897284539951, 2.93913571733158207691707003027, 4.10715158347972286015649299645, 4.75044742748289031222922601625, 5.56498681784481166892228678633, 6.52728041576470757773561459975, 7.22419907225686154066826773454, 8.018368082086072034023308252606, 8.429190100092190501442904145583
