| L(s) = 1 | − 3-s − 4·7-s + 9-s − 3·11-s + 13-s − 7·17-s − 4·19-s + 4·21-s + 7·23-s − 5·25-s − 27-s − 4·29-s + 3·31-s + 3·33-s − 4·37-s − 39-s + 9·41-s + 12·47-s + 9·49-s + 7·51-s + 5·53-s + 4·57-s + 8·61-s − 4·63-s + 7·67-s − 7·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s − 1.69·17-s − 0.917·19-s + 0.872·21-s + 1.45·23-s − 25-s − 0.192·27-s − 0.742·29-s + 0.538·31-s + 0.522·33-s − 0.657·37-s − 0.160·39-s + 1.40·41-s + 1.75·47-s + 9/7·49-s + 0.980·51-s + 0.686·53-s + 0.529·57-s + 1.02·61-s − 0.503·63-s + 0.855·67-s − 0.842·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.231506864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.231506864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−12.61498298743942, −12.31087242938659, −11.43984938355531, −11.16462800454788, −10.72863568701536, −10.40125452014133, −9.827150635229942, −9.315608600700069, −9.054435127196741, −8.492711539741494, −7.930979425421908, −7.314165711108712, −6.800216538970332, −6.602321455186282, −6.083384743504358, −5.485634401184054, −5.218349883380222, −4.384035207838801, −4.018423054016524, −3.566441040886876, −2.722699095786160, −2.440299849921216, −1.863261386725844, −0.7006299687355491, −0.4382777672453961,
0.4382777672453961, 0.7006299687355491, 1.863261386725844, 2.440299849921216, 2.722699095786160, 3.566441040886876, 4.018423054016524, 4.384035207838801, 5.218349883380222, 5.485634401184054, 6.083384743504358, 6.602321455186282, 6.800216538970332, 7.314165711108712, 7.930979425421908, 8.492711539741494, 9.054435127196741, 9.315608600700069, 9.827150635229942, 10.40125452014133, 10.72863568701536, 11.16462800454788, 11.43984938355531, 12.31087242938659, 12.61498298743942
