| L(s) = 1 | + 3-s + 5-s + 3·7-s − 2·9-s + 2·11-s + 13-s + 15-s − 3·17-s + 2·19-s + 3·21-s + 4·23-s − 4·25-s − 5·27-s + 2·29-s + 4·31-s + 2·33-s + 3·35-s + 5·37-s + 39-s − 12·41-s + 7·43-s − 2·45-s − 9·47-s + 2·49-s − 3·51-s + 4·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s + 0.654·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s + 0.371·29-s + 0.718·31-s + 0.348·33-s + 0.507·35-s + 0.821·37-s + 0.160·39-s − 1.87·41-s + 1.06·43-s − 0.298·45-s − 1.31·47-s + 2/7·49-s − 0.420·51-s + 0.549·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.907135495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.907135495\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−11.40591808899451944462613778252, −10.28266601778306219690216338756, −9.190350039528093674482349845769, −8.552472099700084871980246984615, −7.71262456178893161528956236885, −6.48929045010611195636913139276, −5.41207186938958458805928008478, −4.31704488135266103326965750563, −2.91927671703962873131976341826, −1.62873087004997173572398631453,
1.62873087004997173572398631453, 2.91927671703962873131976341826, 4.31704488135266103326965750563, 5.41207186938958458805928008478, 6.48929045010611195636913139276, 7.71262456178893161528956236885, 8.552472099700084871980246984615, 9.190350039528093674482349845769, 10.28266601778306219690216338756, 11.40591808899451944462613778252
