| L(s) = 1 | − 4·7-s + 2·13-s − 8·19-s − 4·31-s + 10·37-s + 8·43-s + 9·49-s − 14·61-s + 16·67-s − 10·73-s + 4·79-s − 8·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.554·13-s − 1.83·19-s − 0.718·31-s + 1.64·37-s + 1.21·43-s + 9/7·49-s − 1.79·61-s + 1.95·67-s − 1.17·73-s + 0.450·79-s − 0.838·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−13.78579561060997, −13.28894209251626, −12.97704343626131, −12.44215268627976, −12.27486492659095, −11.32802025182597, −10.90696462140835, −10.59587269012648, −9.913664625206546, −9.440918538813452, −9.132328307182646, −8.465271719509654, −8.049247260007441, −7.312400660459959, −6.829583165654961, −6.236771497520150, −6.048556599004429, −5.426653912218565, −4.493256542039794, −4.101726135921724, −3.587329830090276, −2.861586466503871, −2.438554103628295, −1.634396447817352, −0.7065225390069753, 0,
0.7065225390069753, 1.634396447817352, 2.438554103628295, 2.861586466503871, 3.587329830090276, 4.101726135921724, 4.493256542039794, 5.426653912218565, 6.048556599004429, 6.236771497520150, 6.829583165654961, 7.312400660459959, 8.049247260007441, 8.465271719509654, 9.132328307182646, 9.440918538813452, 9.913664625206546, 10.59587269012648, 10.90696462140835, 11.32802025182597, 12.27486492659095, 12.44215268627976, 12.97704343626131, 13.28894209251626, 13.78579561060997
