| L(s) = 1 | + 2·2-s + 2·4-s + 4·7-s + 11-s + 6·13-s + 8·14-s − 4·16-s + 17-s − 5·19-s + 2·22-s + 5·23-s + 12·26-s + 8·28-s + 9·29-s + 31-s − 8·32-s + 2·34-s − 10·37-s − 10·38-s + 6·41-s − 6·43-s + 2·44-s + 10·46-s − 10·47-s + 9·49-s + 12·52-s + 9·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.51·7-s + 0.301·11-s + 1.66·13-s + 2.13·14-s − 16-s + 0.242·17-s − 1.14·19-s + 0.426·22-s + 1.04·23-s + 2.35·26-s + 1.51·28-s + 1.67·29-s + 0.179·31-s − 1.41·32-s + 0.342·34-s − 1.64·37-s − 1.62·38-s + 0.937·41-s − 0.914·43-s + 0.301·44-s + 1.47·46-s − 1.45·47-s + 9/7·49-s + 1.66·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.904991048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.904991048\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−8.043494745463270024171163269654, −6.81207129011257344883968023748, −6.53550793665941056708181310168, −5.56020069353088916085008697811, −5.08842499815324580647791531931, −4.35620382740726511841447227378, −3.82650384623874306261078072169, −2.97122059265832012311512255728, −1.96734581475371476691197327577, −1.09126187005683456695519717661,
1.09126187005683456695519717661, 1.96734581475371476691197327577, 2.97122059265832012311512255728, 3.82650384623874306261078072169, 4.35620382740726511841447227378, 5.08842499815324580647791531931, 5.56020069353088916085008697811, 6.53550793665941056708181310168, 6.81207129011257344883968023748, 8.043494745463270024171163269654
