| L(s) = 1 | − 3·5-s − 3·13-s + 3·17-s − 23-s + 4·25-s + 6·29-s + 2·31-s − 4·37-s + 8·41-s + 4·43-s + 11·47-s + 11·53-s + 12·59-s + 12·61-s + 9·65-s − 3·67-s + 5·71-s + 15·73-s + 8·79-s + 12·83-s − 9·85-s − 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.832·13-s + 0.727·17-s − 0.208·23-s + 4/5·25-s + 1.11·29-s + 0.359·31-s − 0.657·37-s + 1.24·41-s + 0.609·43-s + 1.60·47-s + 1.51·53-s + 1.56·59-s + 1.53·61-s + 1.11·65-s − 0.366·67-s + 0.593·71-s + 1.75·73-s + 0.900·79-s + 1.31·83-s − 0.976·85-s − 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.979639472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979639472\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.06701678401506, −13.56652873058442, −12.78965510668241, −12.35018433519310, −11.99404820606669, −11.71744588798440, −10.92232504507248, −10.63989961883129, −9.926573978912229, −9.564561960779375, −8.784689318571381, −8.345322574649211, −7.871060755219778, −7.374339199308473, −6.973501711957104, −6.359151730782057, −5.502924615324052, −5.178675716436557, −4.368257659456018, −3.966788948216950, −3.482993649718513, −2.616522611894537, −2.250814546305415, −0.9455042438166926, −0.5886141418478428,
0.5886141418478428, 0.9455042438166926, 2.250814546305415, 2.616522611894537, 3.482993649718513, 3.966788948216950, 4.368257659456018, 5.178675716436557, 5.502924615324052, 6.359151730782057, 6.973501711957104, 7.374339199308473, 7.871060755219778, 8.345322574649211, 8.784689318571381, 9.564561960779375, 9.926573978912229, 10.63989961883129, 10.92232504507248, 11.71744588798440, 11.99404820606669, 12.35018433519310, 12.78965510668241, 13.56652873058442, 14.06701678401506
