| L(s) = 1 | − 4·7-s − 11-s + 2·13-s + 17-s − 2·29-s + 4·31-s + 6·37-s − 6·41-s − 4·43-s + 9·49-s + 10·53-s + 12·59-s + 2·61-s − 8·67-s − 4·71-s + 6·73-s + 4·77-s − 8·79-s − 4·83-s − 10·89-s − 8·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.301·11-s + 0.554·13-s + 0.242·17-s − 0.371·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 9/7·49-s + 1.37·53-s + 1.56·59-s + 0.256·61-s − 0.977·67-s − 0.474·71-s + 0.702·73-s + 0.455·77-s − 0.900·79-s − 0.439·83-s − 1.05·89-s − 0.838·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.634212072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.634212072\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−12.65751529691689, −12.06722538508247, −11.82885608463217, −11.18898200211522, −10.75207430180327, −10.12049188769839, −9.928267168531774, −9.526131990187304, −8.922839132886284, −8.427086609897459, −8.156604067521447, −7.283580924040510, −7.063750047706985, −6.544276512880929, −5.977639826137052, −5.731231261938825, −5.107907885192286, −4.426525469149808, −3.918594290994949, −3.421456760655966, −2.956376570464921, −2.477140191153270, −1.758879656988030, −0.9590655710815847, −0.3887298244277090,
0.3887298244277090, 0.9590655710815847, 1.758879656988030, 2.477140191153270, 2.956376570464921, 3.421456760655966, 3.918594290994949, 4.426525469149808, 5.107907885192286, 5.731231261938825, 5.977639826137052, 6.544276512880929, 7.063750047706985, 7.283580924040510, 8.156604067521447, 8.427086609897459, 8.922839132886284, 9.526131990187304, 9.928267168531774, 10.12049188769839, 10.75207430180327, 11.18898200211522, 11.82885608463217, 12.06722538508247, 12.65751529691689
