| L(s) = 1 | − 2·3-s + 7-s + 9-s − 6·17-s + 6·19-s − 2·21-s + 23-s − 5·25-s + 4·27-s − 6·29-s + 8·31-s + 2·37-s − 2·41-s − 8·43-s − 8·47-s + 49-s + 12·51-s − 2·53-s − 12·57-s − 6·59-s + 63-s − 12·67-s − 2·69-s − 8·71-s − 6·73-s + 10·75-s − 16·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.45·17-s + 1.37·19-s − 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 1.68·51-s − 0.274·53-s − 1.58·57-s − 0.781·59-s + 0.125·63-s − 1.46·67-s − 0.240·69-s − 0.949·71-s − 0.702·73-s + 1.15·75-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−9.361120626920682552766967596987, −8.429630893089533244710293176268, −7.51686479596460400303178433673, −6.62689974931662635095789969685, −5.89317830408152735471088243694, −5.06992659086327375497431676892, −4.34435790835530441237828655946, −2.98835684759371857772218254259, −1.54541982304235013732105849450, 0,
1.54541982304235013732105849450, 2.98835684759371857772218254259, 4.34435790835530441237828655946, 5.06992659086327375497431676892, 5.89317830408152735471088243694, 6.62689974931662635095789969685, 7.51686479596460400303178433673, 8.429630893089533244710293176268, 9.361120626920682552766967596987
