| L(s) = 1 | + 3-s − 7-s − 2·9-s − 6·11-s − 7·13-s − 17-s − 5·19-s − 21-s − 6·23-s − 5·27-s − 3·29-s + 5·31-s − 6·33-s + 2·37-s − 7·39-s − 6·41-s + 8·43-s − 3·47-s + 49-s − 51-s − 3·53-s − 5·57-s + 3·59-s + 61-s + 2·63-s + 8·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.80·11-s − 1.94·13-s − 0.242·17-s − 1.14·19-s − 0.218·21-s − 1.25·23-s − 0.962·27-s − 0.557·29-s + 0.898·31-s − 1.04·33-s + 0.328·37-s − 1.12·39-s − 0.937·41-s + 1.21·43-s − 0.437·47-s + 1/7·49-s − 0.140·51-s − 0.412·53-s − 0.662·57-s + 0.390·59-s + 0.128·61-s + 0.251·63-s + 0.977·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.44881256711943, −12.74602594487868, −12.48731730471325, −12.18542863711686, −11.31839200719387, −11.04896770095935, −10.42714029679723, −9.824912719985439, −9.785397671690896, −9.144578083200797, −8.325707740209168, −8.112608511001380, −7.852352896179319, −7.083638139961310, −6.729822891319664, −5.991247934652580, −5.429456561217842, −5.143324129736763, −4.418515133562052, −3.967414469025801, −3.153758805030521, −2.548316534639977, −2.428585703453516, −1.869001865235124, −0.4600008556376118, 0,
0.4600008556376118, 1.869001865235124, 2.428585703453516, 2.548316534639977, 3.153758805030521, 3.967414469025801, 4.418515133562052, 5.143324129736763, 5.429456561217842, 5.991247934652580, 6.729822891319664, 7.083638139961310, 7.852352896179319, 8.112608511001380, 8.325707740209168, 9.144578083200797, 9.785397671690896, 9.824912719985439, 10.42714029679723, 11.04896770095935, 11.31839200719387, 12.18542863711686, 12.48731730471325, 12.74602594487868, 13.44881256711943
