| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 9-s − 6·11-s + 2·12-s + 4·13-s − 4·16-s − 7·17-s + 2·18-s + 5·19-s − 12·22-s − 23-s − 5·25-s + 8·26-s + 27-s − 29-s − 4·31-s − 8·32-s − 6·33-s − 14·34-s + 2·36-s − 3·37-s + 10·38-s + 4·39-s − 12·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 1.10·13-s − 16-s − 1.69·17-s + 0.471·18-s + 1.14·19-s − 2.55·22-s − 0.208·23-s − 25-s + 1.56·26-s + 0.192·27-s − 0.185·29-s − 0.718·31-s − 1.41·32-s − 1.04·33-s − 2.40·34-s + 1/3·36-s − 0.493·37-s + 1.62·38-s + 0.640·39-s − 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.00790151752050, −12.44364159302050, −11.97019863894752, −11.33552001201244, −11.12985273523012, −10.58330525641611, −10.15965307907732, −9.467533363489391, −9.064208603027857, −8.622336819118775, −8.080965869911054, −7.623516358338530, −7.166115544010178, −6.593824776473997, −6.101649357091048, −5.607240903288553, −5.148191734175272, −4.837576646464523, −4.103124805119255, −3.712238922208798, −3.360815486032203, −2.648613539676754, −2.250294487393313, −1.867591975674871, −0.7722893013465825, 0,
0.7722893013465825, 1.867591975674871, 2.250294487393313, 2.648613539676754, 3.360815486032203, 3.712238922208798, 4.103124805119255, 4.837576646464523, 5.148191734175272, 5.607240903288553, 6.101649357091048, 6.593824776473997, 7.166115544010178, 7.623516358338530, 8.080965869911054, 8.622336819118775, 9.064208603027857, 9.467533363489391, 10.15965307907732, 10.58330525641611, 11.12985273523012, 11.33552001201244, 11.97019863894752, 12.44364159302050, 13.00790151752050
