| L(s) = 1 | + 3·7-s − 11-s − 6·13-s + 17-s + 19-s + 6·23-s − 5·29-s − 2·31-s − 11·37-s + 9·41-s + 5·47-s + 2·49-s + 3·53-s − 12·59-s + 10·61-s + 4·67-s + 4·71-s − 3·73-s − 3·77-s − 2·79-s − 12·83-s − 4·89-s − 18·91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.301·11-s − 1.66·13-s + 0.242·17-s + 0.229·19-s + 1.25·23-s − 0.928·29-s − 0.359·31-s − 1.80·37-s + 1.40·41-s + 0.729·47-s + 2/7·49-s + 0.412·53-s − 1.56·59-s + 1.28·61-s + 0.488·67-s + 0.474·71-s − 0.351·73-s − 0.341·77-s − 0.225·79-s − 1.31·83-s − 0.423·89-s − 1.88·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−12.95282919648153, −12.56976481169584, −12.27971494343480, −11.64067048877469, −11.29382313225408, −10.78715110154130, −10.44414326363715, −9.811569376959176, −9.356389474217387, −8.980429165949002, −8.346322993629397, −7.898251357559558, −7.393449492640880, −7.126376838427186, −6.632033561484683, −5.638544119933795, −5.330697473779553, −5.103201126256543, −4.346686091937535, −4.042612424427054, −3.083804980683415, −2.757166626197698, −2.014970173137789, −1.597417925589073, −0.7869583070175616, 0,
0.7869583070175616, 1.597417925589073, 2.014970173137789, 2.757166626197698, 3.083804980683415, 4.042612424427054, 4.346686091937535, 5.103201126256543, 5.330697473779553, 5.638544119933795, 6.632033561484683, 7.126376838427186, 7.393449492640880, 7.898251357559558, 8.346322993629397, 8.980429165949002, 9.356389474217387, 9.811569376959176, 10.44414326363715, 10.78715110154130, 11.29382313225408, 11.64067048877469, 12.27971494343480, 12.56976481169584, 12.95282919648153
