| L(s) = 1 | − 2·3-s + 5-s + 9-s − 11-s − 2·15-s − 4·23-s + 25-s + 4·27-s − 2·29-s + 2·31-s + 2·33-s + 6·37-s − 8·41-s + 12·43-s + 45-s + 6·47-s + 6·53-s − 55-s − 10·59-s − 4·61-s + 8·67-s + 8·69-s − 4·71-s + 4·73-s − 2·75-s − 16·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.516·15-s − 0.834·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s + 0.359·31-s + 0.348·33-s + 0.986·37-s − 1.24·41-s + 1.82·43-s + 0.149·45-s + 0.875·47-s + 0.824·53-s − 0.134·55-s − 1.30·59-s − 0.512·61-s + 0.977·67-s + 0.963·69-s − 0.474·71-s + 0.468·73-s − 0.230·75-s − 1.80·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 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
−13.45540222504090, −12.83107008237662, −12.42837103922973, −12.00994315099130, −11.57860988879578, −11.01825072464069, −10.69579858689918, −10.23563061466181, −9.720136041769161, −9.285019083862445, −8.603829986473058, −8.228596708486030, −7.454960161700174, −7.184278796989477, −6.376778024927764, −6.098570018639334, −5.628164743601169, −5.244146258195191, −4.527963906390890, −4.202992876882283, −3.382952615002357, −2.691702262384062, −2.185880897364861, −1.380175681472593, −0.7278830419894913, 0,
0.7278830419894913, 1.380175681472593, 2.185880897364861, 2.691702262384062, 3.382952615002357, 4.202992876882283, 4.527963906390890, 5.244146258195191, 5.628164743601169, 6.098570018639334, 6.376778024927764, 7.184278796989477, 7.454960161700174, 8.228596708486030, 8.603829986473058, 9.285019083862445, 9.720136041769161, 10.23563061466181, 10.69579858689918, 11.01825072464069, 11.57860988879578, 12.00994315099130, 12.42837103922973, 12.83107008237662, 13.45540222504090
