| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s − 3·13-s + 3·15-s + 5·17-s + 6·19-s + 21-s + 4·23-s + 4·25-s − 27-s − 3·29-s + 2·31-s + 3·35-s + 37-s + 3·39-s − 11·41-s − 10·43-s − 3·45-s + 6·47-s + 49-s − 5·51-s − 53-s − 6·57-s + 6·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.832·13-s + 0.774·15-s + 1.21·17-s + 1.37·19-s + 0.218·21-s + 0.834·23-s + 4/5·25-s − 0.192·27-s − 0.557·29-s + 0.359·31-s + 0.507·35-s + 0.164·37-s + 0.480·39-s − 1.71·41-s − 1.52·43-s − 0.447·45-s + 0.875·47-s + 1/7·49-s − 0.700·51-s − 0.137·53-s − 0.794·57-s + 0.781·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−15.08286624049759, −14.78623636648235, −13.83263838375304, −13.57505309759073, −12.68897033264737, −12.32022574665214, −11.82401015617303, −11.60478920173979, −10.95563274976616, −10.25768633388434, −9.861475082514925, −9.304032994079007, −8.576900155547489, −7.879251326063285, −7.529570981026078, −7.043752803441842, −6.494594880275776, −5.593697821239213, −5.139976913289486, −4.661579884423890, −3.773877989804623, −3.358505328129709, −2.789591434900175, −1.587694805278460, −0.7849615055502963, 0,
0.7849615055502963, 1.587694805278460, 2.789591434900175, 3.358505328129709, 3.773877989804623, 4.661579884423890, 5.139976913289486, 5.593697821239213, 6.494594880275776, 7.043752803441842, 7.529570981026078, 7.879251326063285, 8.576900155547489, 9.304032994079007, 9.861475082514925, 10.25768633388434, 10.95563274976616, 11.60478920173979, 11.82401015617303, 12.32022574665214, 12.68897033264737, 13.57505309759073, 13.83263838375304, 14.78623636648235, 15.08286624049759
