| L(s) = 1 | + 7-s + 13-s − 3·17-s − 2·19-s + 3·23-s + 3·29-s + 31-s − 2·37-s − 3·41-s − 7·43-s + 6·47-s + 49-s − 9·53-s − 3·59-s − 61-s + 8·67-s + 4·73-s − 8·79-s + 15·83-s + 6·89-s + 91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.277·13-s − 0.727·17-s − 0.458·19-s + 0.625·23-s + 0.557·29-s + 0.179·31-s − 0.328·37-s − 0.468·41-s − 1.06·43-s + 0.875·47-s + 1/7·49-s − 1.23·53-s − 0.390·59-s − 0.128·61-s + 0.977·67-s + 0.468·73-s − 0.900·79-s + 1.64·83-s + 0.635·89-s + 0.104·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.70123765329846, −14.99394725646458, −14.71066657400773, −13.89884664843782, −13.56082624165348, −13.00315456597972, −12.33792140387202, −11.91464488044562, −11.17469106777404, −10.82307028486489, −10.29655631469324, −9.505014735060567, −9.073424450197107, −8.293672919339425, −8.112413628752682, −7.137593530454838, −6.698842768061830, −6.124986608017662, −5.303204209434167, −4.789613296718812, −4.156725407449832, −3.416849404320872, −2.659074823765597, −1.901994781161181, −1.108766872126341, 0,
1.108766872126341, 1.901994781161181, 2.659074823765597, 3.416849404320872, 4.156725407449832, 4.789613296718812, 5.303204209434167, 6.124986608017662, 6.698842768061830, 7.137593530454838, 8.112413628752682, 8.293672919339425, 9.073424450197107, 9.505014735060567, 10.29655631469324, 10.82307028486489, 11.17469106777404, 11.91464488044562, 12.33792140387202, 13.00315456597972, 13.56082624165348, 13.89884664843782, 14.71066657400773, 14.99394725646458, 15.70123765329846
