| L(s) = 1 | + 3-s + 4·5-s + 2·7-s + 9-s − 2·11-s + 4·15-s + 2·17-s + 6·19-s + 2·21-s + 23-s + 11·25-s + 27-s + 2·29-s + 4·31-s − 2·33-s + 8·35-s − 4·37-s + 6·41-s + 4·45-s − 3·49-s + 2·51-s + 6·53-s − 8·55-s + 6·57-s − 2·61-s + 2·63-s − 14·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.03·15-s + 0.485·17-s + 1.37·19-s + 0.436·21-s + 0.208·23-s + 11/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.348·33-s + 1.35·35-s − 0.657·37-s + 0.937·41-s + 0.596·45-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 1.07·55-s + 0.794·57-s − 0.256·61-s + 0.251·63-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.431328082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.431328082\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.27118282819050, −12.80147946683532, −12.28657679349850, −11.74291981009927, −11.18713273344727, −10.57889622909678, −10.20636975809787, −9.843383714583386, −9.367513748840982, −8.927442001815857, −8.406789658463623, −7.916712886092257, −7.335679695451942, −6.956167956014768, −6.220401501978404, −5.700300656440077, −5.400508550806788, −4.806357175488207, −4.402057524175011, −3.412613590329375, −2.898946453468849, −2.515971606444840, −1.792143818608873, −1.384028350308261, −0.7592802704598119,
0.7592802704598119, 1.384028350308261, 1.792143818608873, 2.515971606444840, 2.898946453468849, 3.412613590329375, 4.402057524175011, 4.806357175488207, 5.400508550806788, 5.700300656440077, 6.220401501978404, 6.956167956014768, 7.335679695451942, 7.916712886092257, 8.406789658463623, 8.927442001815857, 9.367513748840982, 9.843383714583386, 10.20636975809787, 10.57889622909678, 11.18713273344727, 11.74291981009927, 12.28657679349850, 12.80147946683532, 13.27118282819050
