| L(s) = 1 | + 7-s + 4·11-s − 3·13-s − 17-s + 6·19-s + 9·31-s − 4·37-s − 6·41-s + 12·43-s − 10·47-s − 6·49-s − 9·53-s + 14·61-s − 8·67-s − 15·71-s − 12·73-s + 4·77-s − 3·79-s + 6·89-s − 3·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 119-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s − 0.832·13-s − 0.242·17-s + 1.37·19-s + 1.61·31-s − 0.657·37-s − 0.937·41-s + 1.82·43-s − 1.45·47-s − 6/7·49-s − 1.23·53-s + 1.79·61-s − 0.977·67-s − 1.78·71-s − 1.40·73-s + 0.455·77-s − 0.337·79-s + 0.635·89-s − 0.314·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.0916·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.786686064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.786686064\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.06455999853209, −12.19391965408722, −11.89555886993083, −11.57590937984836, −11.24701026261592, −10.42019613413077, −10.04696000672509, −9.695893802817072, −9.051496203927018, −8.822793646747319, −8.118733084945759, −7.688028499226695, −7.194633232967714, −6.752752748097942, −6.189253326686249, −5.775561156452698, −4.955947333686517, −4.750829768195853, −4.211767929426928, −3.447691108586903, −3.087657861391390, −2.414685075231066, −1.655504653937587, −1.243552095839899, −0.4747047111823004,
0.4747047111823004, 1.243552095839899, 1.655504653937587, 2.414685075231066, 3.087657861391390, 3.447691108586903, 4.211767929426928, 4.750829768195853, 4.955947333686517, 5.775561156452698, 6.189253326686249, 6.752752748097942, 7.194633232967714, 7.688028499226695, 8.118733084945759, 8.822793646747319, 9.051496203927018, 9.695893802817072, 10.04696000672509, 10.42019613413077, 11.24701026261592, 11.57590937984836, 11.89555886993083, 12.19391965408722, 13.06455999853209
