| L(s) = 1 | − 3-s + 2·7-s + 9-s − 6·11-s − 13-s − 3·17-s + 5·19-s − 2·21-s − 6·23-s − 27-s + 31-s + 6·33-s + 2·37-s + 39-s − 41-s − 8·43-s − 12·47-s − 3·49-s + 3·51-s + 6·53-s − 5·57-s − 9·59-s + 10·61-s + 2·63-s + 13·67-s + 6·69-s − 15·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s − 0.727·17-s + 1.14·19-s − 0.436·21-s − 1.25·23-s − 0.192·27-s + 0.179·31-s + 1.04·33-s + 0.328·37-s + 0.160·39-s − 0.156·41-s − 1.21·43-s − 1.75·47-s − 3/7·49-s + 0.420·51-s + 0.824·53-s − 0.662·57-s − 1.17·59-s + 1.28·61-s + 0.251·63-s + 1.58·67-s + 0.722·69-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6097125950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6097125950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 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
−13.09592936063344, −12.59093733802650, −12.07977361600597, −11.60231312701811, −11.23402204612270, −10.82765443630203, −10.21752148113386, −9.889664098745795, −9.533300022518032, −8.618327524406690, −8.213392980671843, −7.861366350797632, −7.411605952458520, −6.817267092922118, −6.286871678602965, −5.626819570281108, −5.251213521148387, −4.819612504264517, −4.448139481971794, −3.587312161133676, −3.066273514025922, −2.291593375604132, −1.934697056069274, −1.125926481512409, −0.2386577076299879,
0.2386577076299879, 1.125926481512409, 1.934697056069274, 2.291593375604132, 3.066273514025922, 3.587312161133676, 4.448139481971794, 4.819612504264517, 5.251213521148387, 5.626819570281108, 6.286871678602965, 6.817267092922118, 7.411605952458520, 7.861366350797632, 8.213392980671843, 8.618327524406690, 9.533300022518032, 9.889664098745795, 10.21752148113386, 10.82765443630203, 11.23402204612270, 11.60231312701811, 12.07977361600597, 12.59093733802650, 13.09592936063344
