| L(s) = 1 | − 2·3-s − 2·7-s + 3·9-s + 3·11-s − 13-s − 7·17-s + 6·19-s + 4·21-s − 4·27-s + 8·29-s − 31-s − 6·33-s − 3·37-s + 2·39-s − 11·41-s + 8·43-s − 2·47-s + 3·49-s + 14·51-s + 5·53-s − 12·57-s + 7·59-s − 9·61-s − 6·63-s + 5·67-s + 5·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 9-s + 0.904·11-s − 0.277·13-s − 1.69·17-s + 1.37·19-s + 0.872·21-s − 0.769·27-s + 1.48·29-s − 0.179·31-s − 1.04·33-s − 0.493·37-s + 0.320·39-s − 1.71·41-s + 1.21·43-s − 0.291·47-s + 3/7·49-s + 1.96·51-s + 0.686·53-s − 1.58·57-s + 0.911·59-s − 1.15·61-s − 0.755·63-s + 0.610·67-s + 0.593·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.538533214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.538533214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 72 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 74 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 85 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 92 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 138 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 42 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 21 T + 264 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 140 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.585244150860824686518309520845, −8.371671368790602682572100263770, −7.54836050005079706389827226672, −7.53384080206738098428815992764, −6.84599821124888690530842348777, −6.77755642528178580297004721712, −6.44646081781389674042613170269, −6.11396964975440875906293505678, −5.69072065147117378768568800577, −5.17308645486050380414451766365, −4.91035297721654512964700869398, −4.59935135312921539515821970617, −3.94964176863903294369196888406, −3.81826609164185726572161982238, −3.14823792681983342357937158274, −2.79559883333916263245572892985, −2.06199251561542919930101764602, −1.69377830503567631183511241652, −0.836615271225651063672713237367, −0.51551223731442236977651312402,
0.51551223731442236977651312402, 0.836615271225651063672713237367, 1.69377830503567631183511241652, 2.06199251561542919930101764602, 2.79559883333916263245572892985, 3.14823792681983342357937158274, 3.81826609164185726572161982238, 3.94964176863903294369196888406, 4.59935135312921539515821970617, 4.91035297721654512964700869398, 5.17308645486050380414451766365, 5.69072065147117378768568800577, 6.11396964975440875906293505678, 6.44646081781389674042613170269, 6.77755642528178580297004721712, 6.84599821124888690530842348777, 7.53384080206738098428815992764, 7.54836050005079706389827226672, 8.371671368790602682572100263770, 8.585244150860824686518309520845
