| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 3·9-s + 6·11-s − 3·13-s + 2·15-s + 6·17-s − 4·19-s + 4·21-s − 2·23-s + 2·25-s + 4·27-s − 10·29-s + 16·31-s + 12·33-s + 2·35-s − 4·37-s − 6·39-s + 6·41-s − 43-s + 3·45-s − 2·47-s + 3·49-s + 12·51-s + 11·53-s + 6·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 9-s + 1.80·11-s − 0.832·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s + 0.872·21-s − 0.417·23-s + 2/5·25-s + 0.769·27-s − 1.85·29-s + 2.87·31-s + 2.08·33-s + 0.338·35-s − 0.657·37-s − 0.960·39-s + 0.937·41-s − 0.152·43-s + 0.447·45-s − 0.291·47-s + 3/7·49-s + 1.68·51-s + 1.51·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.017169351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.017169351\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 16 T + 121 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + T + 85 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 75 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 113 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 103 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13 T + 183 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 163 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 261 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + p T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.238256909610463276618380968539, −9.069867361502579942271053463303, −8.658845633830651384768909268310, −8.256683035674711739936586861682, −7.75148335482796847821590972041, −7.66569055159009979782230223228, −7.12230985919452991536831036150, −6.67716728568996542178288816517, −6.11139677632316622368688211140, −6.08838358951654932862855142673, −5.20194676214703834925714109474, −4.93308876280857548302633182269, −4.31560871052560067455718044989, −4.06746777793584371417086405870, −3.50749200502440451111547487037, −3.11749800684678810308801358152, −2.32862714825028516788227863840, −2.10170845931704088501241867159, −1.40192137118289792067054061484, −0.914667651901002173468624619614,
0.914667651901002173468624619614, 1.40192137118289792067054061484, 2.10170845931704088501241867159, 2.32862714825028516788227863840, 3.11749800684678810308801358152, 3.50749200502440451111547487037, 4.06746777793584371417086405870, 4.31560871052560067455718044989, 4.93308876280857548302633182269, 5.20194676214703834925714109474, 6.08838358951654932862855142673, 6.11139677632316622368688211140, 6.67716728568996542178288816517, 7.12230985919452991536831036150, 7.66569055159009979782230223228, 7.75148335482796847821590972041, 8.256683035674711739936586861682, 8.658845633830651384768909268310, 9.069867361502579942271053463303, 9.238256909610463276618380968539
