| L(s) = 1 | + 3-s − 5-s + 9-s − 2·13-s − 15-s − 17-s − 4·19-s + 25-s + 27-s + 10·29-s + 8·31-s − 6·37-s − 2·39-s − 2·41-s + 8·43-s − 45-s − 7·49-s − 51-s + 2·53-s − 4·57-s + 12·59-s + 14·61-s + 2·65-s + 8·67-s − 4·71-s + 6·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 0.242·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.85·29-s + 1.43·31-s − 0.986·37-s − 0.320·39-s − 0.312·41-s + 1.21·43-s − 0.149·45-s − 49-s − 0.140·51-s + 0.274·53-s − 0.529·57-s + 1.56·59-s + 1.79·61-s + 0.248·65-s + 0.977·67-s − 0.474·71-s + 0.702·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.021899228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.021899228\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.342989053294585809262889284091, −7.935107397757558173453054439375, −6.84273108501933370480976161146, −6.56230827993101630949470886209, −5.30447909390898666056860300295, −4.54898604240681793671118516874, −3.87205488544134168783931453301, −2.86858552604654823469183935607, −2.17001602284431169371897012509, −0.78268266080753922519906355482,
0.78268266080753922519906355482, 2.17001602284431169371897012509, 2.86858552604654823469183935607, 3.87205488544134168783931453301, 4.54898604240681793671118516874, 5.30447909390898666056860300295, 6.56230827993101630949470886209, 6.84273108501933370480976161146, 7.935107397757558173453054439375, 8.342989053294585809262889284091
