| L(s) = 1 | − 3-s + 3·7-s − 2·9-s + 2·11-s − 5·17-s + 19-s − 3·21-s + 23-s − 5·25-s + 5·27-s − 3·29-s + 4·31-s − 2·33-s − 2·37-s + 8·41-s + 8·43-s − 8·47-s + 2·49-s + 5·51-s + 9·53-s − 57-s + 59-s + 14·61-s − 6·63-s + 13·67-s − 69-s + 10·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.603·11-s − 1.21·17-s + 0.229·19-s − 0.654·21-s + 0.208·23-s − 25-s + 0.962·27-s − 0.557·29-s + 0.718·31-s − 0.348·33-s − 0.328·37-s + 1.24·41-s + 1.21·43-s − 1.16·47-s + 2/7·49-s + 0.700·51-s + 1.23·53-s − 0.132·57-s + 0.130·59-s + 1.79·61-s − 0.755·63-s + 1.58·67-s − 0.120·69-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.869198948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.869198948\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.62520047340650, −14.01518607209884, −13.50509032711787, −13.02568953860663, −12.19793337482503, −11.86227784380050, −11.40133044289993, −10.91786921604266, −10.72817685451694, −9.709731530225904, −9.332289688839661, −8.714791127725084, −8.105146056530764, −7.852074998166161, −6.811172806241951, −6.660884876413205, −5.788744994107011, −5.385878654993835, −4.833213314395739, −4.135302078483223, −3.719041177773782, −2.580288395577302, −2.183071883106801, −1.277359643682737, −0.5273202700122031,
0.5273202700122031, 1.277359643682737, 2.183071883106801, 2.580288395577302, 3.719041177773782, 4.135302078483223, 4.833213314395739, 5.385878654993835, 5.788744994107011, 6.660884876413205, 6.811172806241951, 7.852074998166161, 8.105146056530764, 8.714791127725084, 9.332289688839661, 9.709731530225904, 10.72817685451694, 10.91786921604266, 11.40133044289993, 11.86227784380050, 12.19793337482503, 13.02568953860663, 13.50509032711787, 14.01518607209884, 14.62520047340650
