| L(s) = 1 | + 2·5-s − 3·7-s + 6·11-s − 5·13-s + 6·17-s + 2·23-s + 3·25-s + 8·29-s + 6·31-s − 6·35-s − 6·37-s + 2·41-s − 11·43-s − 24·47-s + 7·49-s + 24·53-s + 12·55-s − 15·61-s − 10·65-s + 5·67-s − 8·71-s − 10·73-s − 18·77-s − 30·79-s + 16·83-s + 12·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.13·7-s + 1.80·11-s − 1.38·13-s + 1.45·17-s + 0.417·23-s + 3/5·25-s + 1.48·29-s + 1.07·31-s − 1.01·35-s − 0.986·37-s + 0.312·41-s − 1.67·43-s − 3.50·47-s + 49-s + 3.29·53-s + 1.61·55-s − 1.92·61-s − 1.24·65-s + 0.610·67-s − 0.949·71-s − 1.17·73-s − 2.05·77-s − 3.37·79-s + 1.75·83-s + 1.30·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.688337049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.688337049\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−9.292215146999167851860850719654, −8.696437010141285736332475498564, −8.608144310547725187509109870015, −8.207359073457134040592799359705, −7.41224684048805989029681861675, −7.07750475110266814187753928377, −6.97258341629976125252397841656, −6.42760776716733226041916455717, −6.03722422075810834426544526817, −5.90994803859076546520009083673, −5.20176581313010257890585847625, −4.72274050678034133638838704896, −4.59480016405058275990728969613, −3.79989099503595025740514492010, −3.19094196149999124688083490174, −3.16679318095859523051258697062, −2.53656177201499154078334134807, −1.75325302574871283423453290592, −1.35122029720640795778669461058, −0.58480587030308368234404322142,
0.58480587030308368234404322142, 1.35122029720640795778669461058, 1.75325302574871283423453290592, 2.53656177201499154078334134807, 3.16679318095859523051258697062, 3.19094196149999124688083490174, 3.79989099503595025740514492010, 4.59480016405058275990728969613, 4.72274050678034133638838704896, 5.20176581313010257890585847625, 5.90994803859076546520009083673, 6.03722422075810834426544526817, 6.42760776716733226041916455717, 6.97258341629976125252397841656, 7.07750475110266814187753928377, 7.41224684048805989029681861675, 8.207359073457134040592799359705, 8.608144310547725187509109870015, 8.696437010141285736332475498564, 9.292215146999167851860850719654
