| L(s) = 1 | − 2·3-s − 3·5-s − 2·7-s + 9-s − 11-s + 6·15-s − 3·17-s + 2·19-s + 4·21-s + 4·25-s + 4·27-s + 3·29-s − 8·31-s + 2·33-s + 6·35-s − 11·37-s + 9·41-s − 4·43-s − 3·45-s + 6·47-s − 3·49-s + 6·51-s − 3·53-s + 3·55-s − 4·57-s + 6·59-s + 7·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.54·15-s − 0.727·17-s + 0.458·19-s + 0.872·21-s + 4/5·25-s + 0.769·27-s + 0.557·29-s − 1.43·31-s + 0.348·33-s + 1.01·35-s − 1.80·37-s + 1.40·41-s − 0.609·43-s − 0.447·45-s + 0.875·47-s − 3/7·49-s + 0.840·51-s − 0.412·53-s + 0.404·55-s − 0.529·57-s + 0.781·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3929764496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3929764496\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.40378790928182, −12.90136716409191, −12.49988683961271, −12.03881315756334, −11.69914302392891, −11.15419780102426, −10.81194424532215, −10.41166548828084, −9.715332624130803, −9.199074639251122, −8.540663980964131, −8.227093175213191, −7.357354789161485, −7.162115677829695, −6.600338284423612, −6.056518112857303, −5.446243907604622, −5.028767031102771, −4.417271198583893, −3.795721617938262, −3.381207604059612, −2.700054312436672, −1.872381026797986, −0.8231681459281584, −0.2829747301029885,
0.2829747301029885, 0.8231681459281584, 1.872381026797986, 2.700054312436672, 3.381207604059612, 3.795721617938262, 4.417271198583893, 5.028767031102771, 5.446243907604622, 6.056518112857303, 6.600338284423612, 7.162115677829695, 7.357354789161485, 8.227093175213191, 8.540663980964131, 9.199074639251122, 9.715332624130803, 10.41166548828084, 10.81194424532215, 11.15419780102426, 11.69914302392891, 12.03881315756334, 12.49988683961271, 12.90136716409191, 13.40378790928182
