| L(s) = 1 | − 3-s − 5-s − 2·9-s − 3·11-s + 13-s + 15-s + 5·17-s − 6·19-s − 4·23-s + 25-s + 5·27-s + 6·29-s + 7·31-s + 3·33-s − 11·37-s − 39-s + 6·41-s − 4·43-s + 2·45-s + 2·47-s − 5·51-s + 53-s + 3·55-s + 6·57-s + 11·59-s − 65-s + 7·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s + 0.277·13-s + 0.258·15-s + 1.21·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s + 0.962·27-s + 1.11·29-s + 1.25·31-s + 0.522·33-s − 1.80·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s + 0.291·47-s − 0.700·51-s + 0.137·53-s + 0.404·55-s + 0.794·57-s + 1.43·59-s − 0.124·65-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.74762842638820, −12.27723256889578, −11.95299756366205, −11.71172698769431, −10.93548106088473, −10.63662322807595, −10.28636690141805, −9.897850024532389, −9.165093031029789, −8.519532716867235, −8.220805099836614, −8.043844863476531, −7.265759876336578, −6.741057071396136, −6.292787672028045, −5.847550093635374, −5.243953720525417, −5.020260360697092, −4.253961256040512, −3.855290982502420, −3.146511024313485, −2.687987283419861, −2.153090231968030, −1.283973154379122, −0.6138053820497914, 0,
0.6138053820497914, 1.283973154379122, 2.153090231968030, 2.687987283419861, 3.146511024313485, 3.855290982502420, 4.253961256040512, 5.020260360697092, 5.243953720525417, 5.847550093635374, 6.292787672028045, 6.741057071396136, 7.265759876336578, 8.043844863476531, 8.220805099836614, 8.519532716867235, 9.165093031029789, 9.897850024532389, 10.28636690141805, 10.63662322807595, 10.93548106088473, 11.71172698769431, 11.95299756366205, 12.27723256889578, 12.74762842638820
