| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 3·11-s + 6·13-s − 2·14-s + 16-s + 3·17-s + 2·19-s − 3·22-s + 5·23-s − 6·26-s + 2·28-s + 29-s + 4·31-s − 32-s − 3·34-s − 12·37-s − 2·38-s + 12·43-s + 3·44-s − 5·46-s − 4·47-s − 3·49-s + 6·52-s + 4·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.904·11-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s − 0.639·22-s + 1.04·23-s − 1.17·26-s + 0.377·28-s + 0.185·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 1.97·37-s − 0.324·38-s + 1.82·43-s + 0.452·44-s − 0.737·46-s − 0.583·47-s − 3/7·49-s + 0.832·52-s + 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.808761424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808761424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−14.29680253693490, −14.03330195014083, −13.38144992075739, −12.85011510124606, −12.09342744109510, −11.70849347057180, −11.34920236250969, −10.65347391655674, −10.43561852870122, −9.682504242530337, −9.009448268928500, −8.719411756226464, −8.316905603504989, −7.557162338944706, −7.160920390474549, −6.511394083694495, −5.918455126228894, −5.468222845199338, −4.648025066378476, −4.035698433817325, −3.326607644862088, −2.856751050613181, −1.661952804634352, −1.384556320423679, −0.7176351424872421,
0.7176351424872421, 1.384556320423679, 1.661952804634352, 2.856751050613181, 3.326607644862088, 4.035698433817325, 4.648025066378476, 5.468222845199338, 5.918455126228894, 6.511394083694495, 7.160920390474549, 7.557162338944706, 8.316905603504989, 8.719411756226464, 9.009448268928500, 9.682504242530337, 10.43561852870122, 10.65347391655674, 11.34920236250969, 11.70849347057180, 12.09342744109510, 12.85011510124606, 13.38144992075739, 14.03330195014083, 14.29680253693490
