| L(s) = 1 | − 3-s − 3·5-s + 7-s + 9-s − 3·11-s − 5·13-s + 3·15-s + 2·19-s − 21-s + 6·23-s + 4·25-s − 27-s − 6·29-s − 4·31-s + 3·33-s − 3·35-s + 11·37-s + 5·39-s + 12·41-s − 43-s − 3·45-s − 12·47-s + 49-s + 9·53-s + 9·55-s − 2·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.38·13-s + 0.774·15-s + 0.458·19-s − 0.218·21-s + 1.25·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.522·33-s − 0.507·35-s + 1.80·37-s + 0.800·39-s + 1.87·41-s − 0.152·43-s − 0.447·45-s − 1.75·47-s + 1/7·49-s + 1.23·53-s + 1.21·55-s − 0.264·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−12.57271762156802, −12.25553458412310, −11.65768599416323, −11.37037286934360, −10.92802453286502, −10.73568035081475, −9.951304351799266, −9.525733530038576, −9.181497084124552, −8.495297928688423, −7.863472210335002, −7.573457670954945, −7.435310002201619, −6.885873014962444, −6.148164000695609, −5.664499009369479, −5.039987911210806, −4.831101766292344, −4.307151271377241, −3.794568506897518, −3.102169927848824, −2.690466775239625, −2.055121682450859, −1.227441873377578, −0.5464948673780422, 0,
0.5464948673780422, 1.227441873377578, 2.055121682450859, 2.690466775239625, 3.102169927848824, 3.794568506897518, 4.307151271377241, 4.831101766292344, 5.039987911210806, 5.664499009369479, 6.148164000695609, 6.885873014962444, 7.435310002201619, 7.573457670954945, 7.863472210335002, 8.495297928688423, 9.181497084124552, 9.525733530038576, 9.951304351799266, 10.73568035081475, 10.92802453286502, 11.37037286934360, 11.65768599416323, 12.25553458412310, 12.57271762156802
