| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 4·7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 4·14-s + 15-s + 16-s + 2·17-s − 18-s + 3·19-s − 20-s + 4·21-s − 22-s − 3·23-s + 24-s + 25-s − 26-s − 27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.688·19-s − 0.223·20-s + 0.872·21-s − 0.213·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6794879272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6794879272\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 89 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−15.44112533267769, −14.78789164415145, −14.09293623047518, −13.46390072145191, −12.94113965011596, −12.29518059456552, −11.98307385133568, −11.44164526986648, −10.80971433413006, −10.17694690504646, −9.772820327754042, −9.410341192936197, −8.590480915747357, −8.116563787916997, −7.370058894907609, −6.817167802825430, −6.429651135594236, −5.803592449983000, −5.186254815179859, −4.258399830647281, −3.501801090301917, −3.153809264249860, −2.168907252278017, −1.152376636200120, −0.4195678506100672,
0.4195678506100672, 1.152376636200120, 2.168907252278017, 3.153809264249860, 3.501801090301917, 4.258399830647281, 5.186254815179859, 5.803592449983000, 6.429651135594236, 6.817167802825430, 7.370058894907609, 8.116563787916997, 8.590480915747357, 9.410341192936197, 9.772820327754042, 10.17694690504646, 10.80971433413006, 11.44164526986648, 11.98307385133568, 12.29518059456552, 12.94113965011596, 13.46390072145191, 14.09293623047518, 14.78789164415145, 15.44112533267769
