| L(s) = 1 | + 2-s + 4-s + 5-s + 5·7-s + 8-s + 10-s + 11-s + 5·14-s + 16-s − 19-s + 20-s + 22-s + 2·23-s + 25-s + 5·28-s + 4·29-s + 31-s + 32-s + 5·35-s + 4·37-s − 38-s + 40-s + 6·41-s − 43-s + 44-s + 2·46-s + 7·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.88·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.33·14-s + 1/4·16-s − 0.229·19-s + 0.223·20-s + 0.213·22-s + 0.417·23-s + 1/5·25-s + 0.944·28-s + 0.742·29-s + 0.179·31-s + 0.176·32-s + 0.845·35-s + 0.657·37-s − 0.162·38-s + 0.158·40-s + 0.937·41-s − 0.152·43-s + 0.150·44-s + 0.294·46-s + 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.872092987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.872092987\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−13.26564894083638, −12.75299288267866, −12.34253166114289, −11.76999811317590, −11.30260378636856, −11.09389752961089, −10.53487886825397, −10.01772121183041, −9.434149667931428, −8.802828274944136, −8.322954428032118, −7.997551480399143, −7.338946069774298, −6.891858541433398, −6.348959077022197, −5.611277076021892, −5.372279745827187, −4.813446295919277, −4.227600989508711, −4.001281250259024, −3.030276249431500, −2.388365376028565, −2.037735210034545, −1.228524156705658, −0.8418668006740802,
0.8418668006740802, 1.228524156705658, 2.037735210034545, 2.388365376028565, 3.030276249431500, 4.001281250259024, 4.227600989508711, 4.813446295919277, 5.372279745827187, 5.611277076021892, 6.348959077022197, 6.891858541433398, 7.338946069774298, 7.997551480399143, 8.322954428032118, 8.802828274944136, 9.434149667931428, 10.01772121183041, 10.53487886825397, 11.09389752961089, 11.30260378636856, 11.76999811317590, 12.34253166114289, 12.75299288267866, 13.26564894083638
