| L(s) = 1 | + 2·3-s + 3·7-s + 9-s + 4·13-s + 2·17-s − 2·19-s + 6·21-s − 23-s − 4·27-s − 6·29-s + 5·31-s − 8·37-s + 8·39-s − 41-s + 8·43-s − 5·47-s + 2·49-s + 4·51-s + 4·53-s − 4·57-s − 14·59-s + 14·61-s + 3·63-s − 2·67-s − 2·69-s − 8·71-s − 5·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.13·7-s + 1/3·9-s + 1.10·13-s + 0.485·17-s − 0.458·19-s + 1.30·21-s − 0.208·23-s − 0.769·27-s − 1.11·29-s + 0.898·31-s − 1.31·37-s + 1.28·39-s − 0.156·41-s + 1.21·43-s − 0.729·47-s + 2/7·49-s + 0.560·51-s + 0.549·53-s − 0.529·57-s − 1.82·59-s + 1.79·61-s + 0.377·63-s − 0.244·67-s − 0.240·69-s − 0.949·71-s − 0.585·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−13.98298718668124, −13.70001828489690, −13.25477193855866, −12.65650090659412, −12.04074179457540, −11.57482052314923, −11.01080854191043, −10.70261795128166, −10.00926521093402, −9.472799369568235, −8.849469512604335, −8.583825728301948, −8.043199565692294, −7.782581288944584, −7.107040412894151, −6.501777159903015, −5.695823932009279, −5.461712790598311, −4.566692355155588, −4.084218685741134, −3.561742004907476, −2.955759398289782, −2.314955086828564, −1.631144267858379, −1.243287656378754, 0,
1.243287656378754, 1.631144267858379, 2.314955086828564, 2.955759398289782, 3.561742004907476, 4.084218685741134, 4.566692355155588, 5.461712790598311, 5.695823932009279, 6.501777159903015, 7.107040412894151, 7.782581288944584, 8.043199565692294, 8.583825728301948, 8.849469512604335, 9.472799369568235, 10.00926521093402, 10.70261795128166, 11.01080854191043, 11.57482052314923, 12.04074179457540, 12.65650090659412, 13.25477193855866, 13.70001828489690, 13.98298718668124
