| L(s) = 1 | + 3-s − 7-s + 9-s − 2·11-s + 13-s − 6·17-s − 8·19-s − 21-s + 4·23-s − 5·25-s + 27-s + 6·29-s − 4·31-s − 2·33-s + 2·37-s + 39-s + 4·43-s + 6·47-s + 49-s − 6·51-s − 6·53-s − 8·57-s + 10·59-s − 10·61-s − 63-s − 4·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 1.45·17-s − 1.83·19-s − 0.218·21-s + 0.834·23-s − 25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.348·33-s + 0.328·37-s + 0.160·39-s + 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 1.05·57-s + 1.30·59-s − 1.28·61-s − 0.125·63-s − 0.488·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.671490309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.671490309\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−15.70275502983901, −15.32411163543927, −14.88357655783172, −14.11849169303852, −13.61858853774333, −13.02370041784588, −12.78974311791228, −12.08672536734716, −11.20863048750578, −10.76245598847879, −10.34863046252991, −9.492141099526870, −9.026160011727662, −8.480109463989969, −7.974144183558377, −7.192344838235811, −6.585303571335687, −6.115200776631479, −5.240675278076203, −4.377034784789105, −4.058423486450418, −3.068880089202869, −2.412725720598584, −1.831871614159058, −0.5031476356854806,
0.5031476356854806, 1.831871614159058, 2.412725720598584, 3.068880089202869, 4.058423486450418, 4.377034784789105, 5.240675278076203, 6.115200776631479, 6.585303571335687, 7.192344838235811, 7.974144183558377, 8.480109463989969, 9.026160011727662, 9.492141099526870, 10.34863046252991, 10.76245598847879, 11.20863048750578, 12.08672536734716, 12.78974311791228, 13.02370041784588, 13.61858853774333, 14.11849169303852, 14.88357655783172, 15.32411163543927, 15.70275502983901
