L(s) = 1 | + 7-s − 3·9-s + 4·11-s + 2·13-s + 4·17-s − 4·19-s − 5·25-s − 2·29-s + 4·31-s − 4·37-s − 6·41-s + 4·43-s + 4·47-s + 49-s − 12·53-s − 8·59-s − 3·63-s − 4·67-s + 8·71-s − 10·73-s + 4·77-s − 8·79-s + 9·81-s + 12·83-s − 4·89-s + 2·91-s + 12·97-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 9-s + 1.20·11-s + 0.554·13-s + 0.970·17-s − 0.917·19-s − 25-s − 0.371·29-s + 0.718·31-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 1.64·53-s − 1.04·59-s − 0.377·63-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.455·77-s − 0.900·79-s + 81-s + 1.31·83-s − 0.423·89-s + 0.209·91-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.19720287603442, −12.49045043701971, −12.19272920578576, −11.65352661065101, −11.44050352966906, −10.82517352814053, −10.45058198706354, −9.859426581431601, −9.250759382192965, −9.010967793525084, −8.333999353845320, −8.135667091263667, −7.524899929093115, −6.942315018217133, −6.278840310121873, −6.050283007716587, −5.570253790903216, −4.895235492538888, −4.368679337960943, −3.787507852992956, −3.349385512049100, −2.799514788502456, −1.937991269931837, −1.578118990442126, −0.8127673865331493, 0,
0.8127673865331493, 1.578118990442126, 1.937991269931837, 2.799514788502456, 3.349385512049100, 3.787507852992956, 4.368679337960943, 4.895235492538888, 5.570253790903216, 6.050283007716587, 6.278840310121873, 6.942315018217133, 7.524899929093115, 8.135667091263667, 8.333999353845320, 9.010967793525084, 9.250759382192965, 9.859426581431601, 10.45058198706354, 10.82517352814053, 11.44050352966906, 11.65352661065101, 12.19272920578576, 12.49045043701971, 13.19720287603442