| L(s) = 1 | + 3-s + 7-s + 9-s + 2·11-s − 4·13-s − 6·17-s + 6·19-s + 21-s + 8·23-s + 27-s − 2·29-s + 10·31-s + 2·33-s − 2·37-s − 4·39-s + 10·41-s + 4·43-s + 8·47-s + 49-s − 6·51-s − 4·53-s + 6·57-s − 8·59-s + 6·61-s + 63-s − 12·67-s + 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 1.45·17-s + 1.37·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s − 0.371·29-s + 1.79·31-s + 0.348·33-s − 0.328·37-s − 0.640·39-s + 1.56·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.549·53-s + 0.794·57-s − 1.04·59-s + 0.768·61-s + 0.125·63-s − 1.46·67-s + 0.963·69-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
| \(L(1)\) |
\(\approx\) |
\(2.360450138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360450138\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−9.195344474993668828491819123627, −8.418885419584950779629768276168, −7.41457982364605622153290477276, −7.05541844049590231106484265310, −5.97908826020656530634665848482, −4.85556732460565523370416594711, −4.35295828278136936402517968941, −3.09662479856183263181704166983, −2.34661437474864053134360607230, −1.03764982844554314606947876404,
1.03764982844554314606947876404, 2.34661437474864053134360607230, 3.09662479856183263181704166983, 4.35295828278136936402517968941, 4.85556732460565523370416594711, 5.97908826020656530634665848482, 7.05541844049590231106484265310, 7.41457982364605622153290477276, 8.418885419584950779629768276168, 9.195344474993668828491819123627
