| L(s) = 1 | − 3-s − 4·7-s + 9-s + 2·13-s − 6·17-s − 4·19-s + 4·21-s − 27-s + 6·29-s − 8·31-s + 2·37-s − 2·39-s − 6·41-s + 4·43-s + 9·49-s + 6·51-s − 6·53-s + 4·57-s + 10·61-s − 4·63-s + 4·67-s − 2·73-s − 8·79-s + 81-s − 12·83-s − 6·87-s + 18·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.872·21-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 0.529·57-s + 1.28·61-s − 0.503·63-s + 0.488·67-s − 0.234·73-s − 0.900·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7326314990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7326314990\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−8.468240536036555704630848503198, −7.31357774249742150991297991120, −6.59910634542849161459156625156, −6.32472569150928839317323168550, −5.49712735041431211178430480723, −4.49368181982186146386252774551, −3.82348229275525279836763542718, −2.92788308932045914068085173884, −1.92466750882191143426507910728, −0.46487860020891297578644501903,
0.46487860020891297578644501903, 1.92466750882191143426507910728, 2.92788308932045914068085173884, 3.82348229275525279836763542718, 4.49368181982186146386252774551, 5.49712735041431211178430480723, 6.32472569150928839317323168550, 6.59910634542849161459156625156, 7.31357774249742150991297991120, 8.468240536036555704630848503198
