L(s) = 1 | + 2-s + 3·3-s + 4-s + 3·6-s + 7-s + 8-s + 6·9-s + 2·11-s + 3·12-s + 14-s + 16-s + 3·17-s + 6·18-s − 6·19-s + 3·21-s + 2·22-s + 4·23-s + 3·24-s + 9·27-s + 28-s + 2·29-s − 4·31-s + 32-s + 6·33-s + 3·34-s + 6·36-s + 3·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.603·11-s + 0.866·12-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 1.41·18-s − 1.37·19-s + 0.654·21-s + 0.426·22-s + 0.834·23-s + 0.612·24-s + 1.73·27-s + 0.188·28-s + 0.371·29-s − 0.718·31-s + 0.176·32-s + 1.04·33-s + 0.514·34-s + 36-s + 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.548084687\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.548084687\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.78618160341898961599148312743, −7.23568791400333437059684221061, −6.51632105010820945514209338703, −5.69499349255850374306805786509, −4.65916643268043458244706899373, −4.15748343893950010861474305754, −3.45971551695615743734182930904, −2.76910930427420910776332481273, −2.04502161572044176240278653055, −1.23311319217689188134168770571,
1.23311319217689188134168770571, 2.04502161572044176240278653055, 2.76910930427420910776332481273, 3.45971551695615743734182930904, 4.15748343893950010861474305754, 4.65916643268043458244706899373, 5.69499349255850374306805786509, 6.51632105010820945514209338703, 7.23568791400333437059684221061, 7.78618160341898961599148312743