| L(s) = 1 | − 3-s + 7-s − 2·9-s + 11-s + 13-s + 17-s − 3·19-s − 21-s + 6·23-s + 5·27-s − 29-s + 7·31-s − 33-s − 7·37-s − 39-s + 6·41-s + 4·43-s + 6·47-s − 6·49-s − 51-s + 5·53-s + 3·57-s + 14·59-s + 13·61-s − 2·63-s + 8·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.242·17-s − 0.688·19-s − 0.218·21-s + 1.25·23-s + 0.962·27-s − 0.185·29-s + 1.25·31-s − 0.174·33-s − 1.15·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.875·47-s − 6/7·49-s − 0.140·51-s + 0.686·53-s + 0.397·57-s + 1.82·59-s + 1.66·61-s − 0.251·63-s + 0.977·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.143084110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.143084110\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 17 T + p T^{2} \) | 1.89.ar |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.28221425935516, −14.08342840912231, −13.20239779453293, −12.88446822576921, −12.26331818143039, −11.73996161488536, −11.23381487671707, −11.04230438540301, −10.22725497554619, −9.968727454387421, −8.995989923628461, −8.680299003194801, −8.304001506342073, −7.456694442385926, −6.987903917372886, −6.352717548646543, −5.910066492286749, −5.235715708093204, −4.873186187429539, −4.093590019300665, −3.519044948294198, −2.703152980358712, −2.167879436508948, −1.118909720863014, −0.6056077806650433,
0.6056077806650433, 1.118909720863014, 2.167879436508948, 2.703152980358712, 3.519044948294198, 4.093590019300665, 4.873186187429539, 5.235715708093204, 5.910066492286749, 6.352717548646543, 6.987903917372886, 7.456694442385926, 8.304001506342073, 8.680299003194801, 8.995989923628461, 9.968727454387421, 10.22725497554619, 11.04230438540301, 11.23381487671707, 11.73996161488536, 12.26331818143039, 12.88446822576921, 13.20239779453293, 14.08342840912231, 14.28221425935516
