| L(s) = 1 | − 3·11-s + 13-s − 3·17-s − 5·19-s − 5·25-s + 9·29-s − 8·31-s + 2·37-s + 6·41-s − 8·43-s − 9·47-s + 3·53-s + 9·59-s − 7·61-s − 5·67-s + 3·71-s + 2·73-s + 10·79-s − 12·83-s − 6·89-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.904·11-s + 0.277·13-s − 0.727·17-s − 1.14·19-s − 25-s + 1.67·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s − 1.21·43-s − 1.31·47-s + 0.412·53-s + 1.17·59-s − 0.896·61-s − 0.610·67-s + 0.356·71-s + 0.234·73-s + 1.12·79-s − 1.31·83-s − 0.635·89-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7716300416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7716300416\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.76426331128524, −13.30123759772299, −12.92066108608494, −12.48489302636371, −11.87578745273371, −11.26805075160508, −10.94047991507828, −10.38559340000755, −9.952596042216162, −9.392029953481479, −8.748372167733615, −8.277208530984160, −7.982055782254096, −7.179352112716421, −6.780236948565037, −6.112660545081308, −5.739978500222985, −4.950685874743422, −4.563181034726870, −3.909495765447309, −3.304907402142994, −2.527658990559923, −2.105947727113746, −1.329538049410281, −0.2743383268750708,
0.2743383268750708, 1.329538049410281, 2.105947727113746, 2.527658990559923, 3.304907402142994, 3.909495765447309, 4.563181034726870, 4.950685874743422, 5.739978500222985, 6.112660545081308, 6.780236948565037, 7.179352112716421, 7.982055782254096, 8.277208530984160, 8.748372167733615, 9.392029953481479, 9.952596042216162, 10.38559340000755, 10.94047991507828, 11.26805075160508, 11.87578745273371, 12.48489302636371, 12.92066108608494, 13.30123759772299, 13.76426331128524
