| L(s) = 1 | + 2-s + 2·3-s + 4-s − 5-s + 2·6-s + 8-s + 9-s − 10-s − 5·11-s + 2·12-s + 13-s − 2·15-s + 16-s − 17-s + 18-s + 3·19-s − 20-s − 5·22-s + 7·23-s + 2·24-s + 25-s + 26-s − 4·27-s − 2·29-s − 2·30-s + 32-s − 10·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.577·12-s + 0.277·13-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.688·19-s − 0.223·20-s − 1.06·22-s + 1.45·23-s + 0.408·24-s + 1/5·25-s + 0.196·26-s − 0.769·27-s − 0.371·29-s − 0.365·30-s + 0.176·32-s − 1.74·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.365551010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.365551010\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81194905020727084935013725602, −7.35917900829736320360424374855, −6.42142687633780071611374217328, −5.55427928050663616725523378816, −4.96683348490974095539738901714, −4.14095418373686422546281131002, −3.38670772128840473483603551547, −2.74555017087184835790545902374, −2.27702993700563300759591554228, −0.857400564408475905491731723279,
0.857400564408475905491731723279, 2.27702993700563300759591554228, 2.74555017087184835790545902374, 3.38670772128840473483603551547, 4.14095418373686422546281131002, 4.96683348490974095539738901714, 5.55427928050663616725523378816, 6.42142687633780071611374217328, 7.35917900829736320360424374855, 7.81194905020727084935013725602
