| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 2·13-s + 15-s + 2·17-s − 4·19-s + 21-s + 25-s − 27-s + 2·29-s − 4·31-s + 35-s + 2·37-s − 2·39-s − 2·41-s − 8·43-s − 45-s + 4·47-s + 49-s − 2·51-s − 2·53-s + 4·57-s − 12·59-s − 14·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s − 0.320·39-s − 0.312·41-s − 1.21·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.529·57-s − 1.56·59-s − 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 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 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.942471236689840715697509513734, −8.151651829084406153563307651295, −7.30726263752797309449269528432, −6.46758655181152769398156304643, −5.81395698986086712659340393304, −4.78820327837737145582447390742, −3.94287049628736653860266238073, −2.98622062715698378788566059984, −1.50269272746963719654878484991, 0,
1.50269272746963719654878484991, 2.98622062715698378788566059984, 3.94287049628736653860266238073, 4.78820327837737145582447390742, 5.81395698986086712659340393304, 6.46758655181152769398156304643, 7.30726263752797309449269528432, 8.151651829084406153563307651295, 8.942471236689840715697509513734
