| L(s) = 1 | + 4·7-s − 4·11-s − 13-s + 2·17-s + 8·19-s − 6·29-s + 4·31-s + 2·37-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s − 10·53-s + 4·59-s − 2·61-s − 16·67-s − 8·71-s − 2·73-s − 16·77-s − 8·79-s − 12·83-s − 14·89-s − 4·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s − 0.277·13-s + 0.485·17-s + 1.83·19-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s + 0.520·59-s − 0.256·61-s − 1.95·67-s − 0.949·71-s − 0.234·73-s − 1.82·77-s − 0.900·79-s − 1.31·83-s − 1.48·89-s − 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 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 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.78150981510082, −14.33207033752309, −13.95794353751686, −13.33151228156075, −12.84503381427301, −12.22508725778471, −11.66586130516652, −11.22097355852391, −10.89691057712667, −10.09199024889698, −9.761456358082852, −9.083048128148338, −8.431776746274967, −7.765669496041967, −7.632496813922926, −7.173864730491024, −6.028754383056255, −5.618127862626301, −5.059845841449695, −4.643091966861216, −3.944280097109456, −2.955459565065836, −2.650537586404838, −1.604992353444383, −1.169671565968840, 0,
1.169671565968840, 1.604992353444383, 2.650537586404838, 2.955459565065836, 3.944280097109456, 4.643091966861216, 5.059845841449695, 5.618127862626301, 6.028754383056255, 7.173864730491024, 7.632496813922926, 7.765669496041967, 8.431776746274967, 9.083048128148338, 9.761456358082852, 10.09199024889698, 10.89691057712667, 11.22097355852391, 11.66586130516652, 12.22508725778471, 12.84503381427301, 13.33151228156075, 13.95794353751686, 14.33207033752309, 14.78150981510082
