| L(s) = 1 | − 3-s − 7-s + 9-s + 6·11-s − 2·13-s − 2·17-s + 19-s + 21-s + 2·23-s − 27-s − 6·33-s + 2·37-s + 2·39-s + 6·47-s + 49-s + 2·51-s − 8·53-s − 57-s + 10·61-s − 63-s + 4·67-s − 2·69-s + 10·73-s − 6·77-s + 8·79-s + 81-s − 14·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.485·17-s + 0.229·19-s + 0.218·21-s + 0.417·23-s − 0.192·27-s − 1.04·33-s + 0.328·37-s + 0.320·39-s + 0.875·47-s + 1/7·49-s + 0.280·51-s − 1.09·53-s − 0.132·57-s + 1.28·61-s − 0.125·63-s + 0.488·67-s − 0.240·69-s + 1.17·73-s − 0.683·77-s + 0.900·79-s + 1/9·81-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.55195120826034, −12.92559213855254, −12.42929477838178, −12.17295542350125, −11.57900026043731, −11.20262470086377, −10.84291235675977, −10.03416360903303, −9.748141358116359, −9.211258049429689, −8.900611015307622, −8.221950877843694, −7.604287893770765, −6.987412435905030, −6.662147746871807, −6.304590140914639, −5.632000603807300, −5.140514895516202, −4.494875642020124, −4.006797422561022, −3.577601855961184, −2.795545391004531, −2.159592972179653, −1.377646145080891, −0.8718078946136790, 0,
0.8718078946136790, 1.377646145080891, 2.159592972179653, 2.795545391004531, 3.577601855961184, 4.006797422561022, 4.494875642020124, 5.140514895516202, 5.632000603807300, 6.304590140914639, 6.662147746871807, 6.987412435905030, 7.604287893770765, 8.221950877843694, 8.900611015307622, 9.211258049429689, 9.748141358116359, 10.03416360903303, 10.84291235675977, 11.20262470086377, 11.57900026043731, 12.17295542350125, 12.42929477838178, 12.92559213855254, 13.55195120826034
