| L(s) = 1 | − 3-s + 9-s + 5·11-s + 2·13-s + 6·17-s + 2·19-s + 5·23-s − 27-s + 5·29-s − 4·31-s − 5·33-s − 37-s − 2·39-s − 12·41-s − 5·43-s + 2·47-s − 6·51-s − 14·53-s − 2·57-s − 2·59-s + 5·67-s − 5·69-s − 9·71-s + 10·73-s + 11·79-s + 81-s − 16·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.50·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s + 1.04·23-s − 0.192·27-s + 0.928·29-s − 0.718·31-s − 0.870·33-s − 0.164·37-s − 0.320·39-s − 1.87·41-s − 0.762·43-s + 0.291·47-s − 0.840·51-s − 1.92·53-s − 0.264·57-s − 0.260·59-s + 0.610·67-s − 0.601·69-s − 1.06·71-s + 1.17·73-s + 1.23·79-s + 1/9·81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.03174433798604, −12.52131730583995, −12.23308744291412, −11.62850453352819, −11.48519579774029, −10.85344908532758, −10.41482789153604, −9.802255911475496, −9.531115183791957, −8.985822479150967, −8.416043890631437, −8.050313740474269, −7.289146951403513, −6.933146459658312, −6.441959998339643, −6.069297855633488, −5.321764415273027, −5.107263346059022, −4.412791339177432, −3.796380453956155, −3.325620456764769, −2.955440756291137, −1.822236395983410, −1.322216691286323, −0.9970054587690613, 0,
0.9970054587690613, 1.322216691286323, 1.822236395983410, 2.955440756291137, 3.325620456764769, 3.796380453956155, 4.412791339177432, 5.107263346059022, 5.321764415273027, 6.069297855633488, 6.441959998339643, 6.933146459658312, 7.289146951403513, 8.050313740474269, 8.416043890631437, 8.985822479150967, 9.531115183791957, 9.802255911475496, 10.41482789153604, 10.85344908532758, 11.48519579774029, 11.62850453352819, 12.23308744291412, 12.52131730583995, 13.03174433798604
