| L(s) = 1 | − 3-s − 4·5-s − 7-s + 9-s + 2·11-s − 2·13-s + 4·15-s + 4·19-s + 21-s + 6·23-s + 11·25-s − 27-s − 10·29-s + 8·31-s − 2·33-s + 4·35-s + 10·37-s + 2·39-s − 4·41-s + 8·43-s − 4·45-s + 4·47-s + 49-s + 10·53-s − 8·55-s − 4·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 1.03·15-s + 0.917·19-s + 0.218·21-s + 1.25·23-s + 11/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.348·33-s + 0.676·35-s + 1.64·37-s + 0.320·39-s − 0.624·41-s + 1.21·43-s − 0.596·45-s + 0.583·47-s + 1/7·49-s + 1.37·53-s − 1.07·55-s − 0.529·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7784733987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7784733987\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−10.77656405250491592073460725453, −9.613256328671100847946136436703, −8.790884401826029462010316690350, −7.53935193216942884829954158220, −7.27987334109443228148867323301, −6.07877945965578593249857952220, −4.83790832179161095688226596322, −4.03196371863461298187155314836, −3.02829428846360456459960342661, −0.77310464999562111477653259452,
0.77310464999562111477653259452, 3.02829428846360456459960342661, 4.03196371863461298187155314836, 4.83790832179161095688226596322, 6.07877945965578593249857952220, 7.27987334109443228148867323301, 7.53935193216942884829954158220, 8.790884401826029462010316690350, 9.613256328671100847946136436703, 10.77656405250491592073460725453
