| L(s) = 1 | − 11-s + 6·13-s − 2·19-s + 4·23-s − 5·25-s + 2·29-s + 2·31-s + 2·37-s − 8·41-s + 2·47-s + 10·53-s + 4·59-s − 10·61-s − 4·67-s + 8·73-s − 8·79-s + 2·83-s − 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 1.66·13-s − 0.458·19-s + 0.834·23-s − 25-s + 0.371·29-s + 0.359·31-s + 0.328·37-s − 1.24·41-s + 0.291·47-s + 1.37·53-s + 0.520·59-s − 1.28·61-s − 0.488·67-s + 0.936·73-s − 0.900·79-s + 0.219·83-s − 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.11292282683322, −13.70414351636360, −13.32817634428556, −12.93504888645507, −12.25153873337423, −11.76176901837109, −11.23649301332942, −10.84030919112377, −10.25437058036813, −9.902752082054387, −8.992916545850311, −8.819619924410196, −8.191647790931465, −7.750916146546209, −7.043282198282612, −6.482178758395823, −6.056505185356032, −5.457252119708633, −4.901126613036559, −4.106024541547620, −3.756113124282459, −3.030847910117472, −2.407609568793354, −1.564297789633255, −1.002724997020092, 0,
1.002724997020092, 1.564297789633255, 2.407609568793354, 3.030847910117472, 3.756113124282459, 4.106024541547620, 4.901126613036559, 5.457252119708633, 6.056505185356032, 6.482178758395823, 7.043282198282612, 7.750916146546209, 8.191647790931465, 8.819619924410196, 8.992916545850311, 9.902752082054387, 10.25437058036813, 10.84030919112377, 11.23649301332942, 11.76176901837109, 12.25153873337423, 12.93504888645507, 13.32817634428556, 13.70414351636360, 14.11292282683322
