| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 11-s + 16-s + 2·17-s + 2·19-s + 20-s + 22-s + 6·23-s + 25-s + 2·29-s + 32-s + 2·34-s − 6·37-s + 2·38-s + 40-s + 2·41-s − 2·43-s + 44-s + 6·46-s − 2·47-s − 7·49-s + 50-s + 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 1/4·16-s + 0.485·17-s + 0.458·19-s + 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.371·29-s + 0.176·32-s + 0.342·34-s − 0.986·37-s + 0.324·38-s + 0.158·40-s + 0.312·41-s − 0.304·43-s + 0.150·44-s + 0.884·46-s − 0.291·47-s − 49-s + 0.141·50-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.806077351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.806077351\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−10.05693826052289398895973233492, −9.238630209310249494130462670649, −8.296093105078622279432498011535, −7.22975619619955006036786500216, −6.53672647440010723921175950694, −5.53973883092288759416369736031, −4.85348839068468109260104334820, −3.66813274923450053298536804164, −2.72485757784841090142318569805, −1.36233205233619760868374974079,
1.36233205233619760868374974079, 2.72485757784841090142318569805, 3.66813274923450053298536804164, 4.85348839068468109260104334820, 5.53973883092288759416369736031, 6.53672647440010723921175950694, 7.22975619619955006036786500216, 8.296093105078622279432498011535, 9.238630209310249494130462670649, 10.05693826052289398895973233492
