| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 2·13-s + 15-s + 4·19-s − 4·21-s + 25-s + 27-s + 6·29-s + 8·31-s − 4·35-s − 2·37-s + 2·39-s + 6·41-s + 4·43-s + 45-s + 9·49-s − 6·53-s + 4·57-s + 10·61-s − 4·63-s + 2·65-s + 4·67-s − 2·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.917·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.824·53-s + 0.529·57-s + 1.28·61-s − 0.503·63-s + 0.248·65-s + 0.488·67-s − 0.234·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.371416257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.371416257\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.11433590456015, −13.62648687873405, −13.22958291556172, −12.73226661873990, −12.31451290738799, −11.71851441685929, −11.10551514674242, −10.37716167882628, −9.987195619216976, −9.683351798503603, −9.030626601706804, −8.710860171281177, −7.992734008371133, −7.455603481152782, −6.796456497374912, −6.321901734253546, −6.000727662182286, −5.190952857628411, −4.570014322638443, −3.800596636511722, −3.320517916957984, −2.768372300803544, −2.280429332860410, −1.213639792480949, −0.6472613868858427,
0.6472613868858427, 1.213639792480949, 2.280429332860410, 2.768372300803544, 3.320517916957984, 3.800596636511722, 4.570014322638443, 5.190952857628411, 6.000727662182286, 6.321901734253546, 6.796456497374912, 7.455603481152782, 7.992734008371133, 8.710860171281177, 9.030626601706804, 9.683351798503603, 9.987195619216976, 10.37716167882628, 11.10551514674242, 11.71851441685929, 12.31451290738799, 12.73226661873990, 13.22958291556172, 13.62648687873405, 14.11433590456015
