| L(s) = 1 | + 3-s + 9-s + 2·13-s − 6·17-s + 4·23-s + 27-s + 2·29-s + 8·31-s + 6·37-s + 2·39-s + 6·41-s + 12·43-s − 12·47-s − 6·51-s − 10·53-s + 8·59-s − 10·61-s − 12·67-s + 4·69-s + 8·71-s + 10·73-s + 16·79-s + 81-s − 12·83-s + 2·87-s + 6·89-s + 8·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.554·13-s − 1.45·17-s + 0.834·23-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.320·39-s + 0.937·41-s + 1.82·43-s − 1.75·47-s − 0.840·51-s − 1.37·53-s + 1.04·59-s − 1.28·61-s − 1.46·67-s + 0.481·69-s + 0.949·71-s + 1.17·73-s + 1.80·79-s + 1/9·81-s − 1.31·83-s + 0.214·87-s + 0.635·89-s + 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.900451485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.900451485\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.08449509273666, −12.57292303596105, −12.00035769349762, −11.42553555093653, −11.00306258084539, −10.71518290178592, −10.09335477543089, −9.430715008857295, −9.235698380848821, −8.745123384495195, −8.117054961460144, −7.913088644626407, −7.231043694325293, −6.700542342945368, −6.223052003312187, −5.942006833281602, −4.907944918405787, −4.663829223397616, −4.191105601836482, −3.499621169168143, −2.959038869239825, −2.459819578156191, −1.918235272991480, −1.109885376343515, −0.5714857861629622,
0.5714857861629622, 1.109885376343515, 1.918235272991480, 2.459819578156191, 2.959038869239825, 3.499621169168143, 4.191105601836482, 4.663829223397616, 4.907944918405787, 5.942006833281602, 6.223052003312187, 6.700542342945368, 7.231043694325293, 7.913088644626407, 8.117054961460144, 8.745123384495195, 9.235698380848821, 9.430715008857295, 10.09335477543089, 10.71518290178592, 11.00306258084539, 11.42553555093653, 12.00035769349762, 12.57292303596105, 13.08449509273666
