| L(s) = 1 | + 5-s + 4·11-s + 2·13-s − 6·17-s − 2·19-s + 25-s + 6·29-s − 2·31-s + 37-s − 8·41-s − 6·43-s − 12·47-s − 7·49-s + 4·55-s + 12·59-s − 10·61-s + 2·65-s + 4·67-s − 4·71-s + 6·73-s − 10·79-s − 6·85-s − 10·89-s − 2·95-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.458·19-s + 1/5·25-s + 1.11·29-s − 0.359·31-s + 0.164·37-s − 1.24·41-s − 0.914·43-s − 1.75·47-s − 49-s + 0.539·55-s + 1.56·59-s − 1.28·61-s + 0.248·65-s + 0.488·67-s − 0.474·71-s + 0.702·73-s − 1.12·79-s − 0.650·85-s − 1.05·89-s − 0.205·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.00149955790978, −13.27520576067710, −13.17073931232546, −12.53376781590493, −11.90158534758570, −11.41599705415810, −11.16011891848520, −10.49570288243358, −9.898407277085620, −9.615923158756745, −8.860184931258860, −8.521812222265788, −8.268407377130150, −7.208767971869398, −6.845449794692886, −6.277286472133995, −6.143471652364540, −5.180057464184555, −4.653562405945945, −4.249177711872579, −3.444584152194446, −3.074191470195749, −2.011372205329220, −1.786295151226958, −0.9421184780929796, 0,
0.9421184780929796, 1.786295151226958, 2.011372205329220, 3.074191470195749, 3.444584152194446, 4.249177711872579, 4.653562405945945, 5.180057464184555, 6.143471652364540, 6.277286472133995, 6.845449794692886, 7.208767971869398, 8.268407377130150, 8.521812222265788, 8.860184931258860, 9.615923158756745, 9.898407277085620, 10.49570288243358, 11.16011891848520, 11.41599705415810, 11.90158534758570, 12.53376781590493, 13.17073931232546, 13.27520576067710, 14.00149955790978
