| L(s) = 1 | + 2·3-s + 5-s + 7-s + 9-s + 2·11-s + 2·15-s − 3·17-s − 8·19-s + 2·21-s + 6·23-s − 4·25-s − 4·27-s − 9·29-s − 6·31-s + 4·33-s + 35-s + 3·37-s − 3·41-s − 2·43-s + 45-s − 12·47-s + 49-s − 6·51-s − 5·53-s + 2·55-s − 16·57-s − 14·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.516·15-s − 0.727·17-s − 1.83·19-s + 0.436·21-s + 1.25·23-s − 4/5·25-s − 0.769·27-s − 1.67·29-s − 1.07·31-s + 0.696·33-s + 0.169·35-s + 0.493·37-s − 0.468·41-s − 0.304·43-s + 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.840·51-s − 0.686·53-s + 0.269·55-s − 2.11·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.58317621110154684086312461226, −6.63054341853485842598907948149, −6.16877732005839014787967296401, −5.22145286760892230184596343896, −4.44167050893745904133587596855, −3.72362909357795053617479780648, −3.04211303950355410822456397532, −1.92540472733853759399881368346, −1.82427496262827446013758055637, 0,
1.82427496262827446013758055637, 1.92540472733853759399881368346, 3.04211303950355410822456397532, 3.72362909357795053617479780648, 4.44167050893745904133587596855, 5.22145286760892230184596343896, 6.16877732005839014787967296401, 6.63054341853485842598907948149, 7.58317621110154684086312461226
