| L(s) = 1 | − 2·9-s + 4·13-s + 4·29-s − 4·31-s − 4·59-s + 3·81-s − 8·117-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 2·9-s + 4·13-s + 4·29-s − 4·31-s − 4·59-s + 3·81-s − 8·117-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.327646401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327646401\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.30781155296319202107101683037, −6.21124507125463421976822979471, −5.97057968165574326904486221539, −5.94605151962651728062059924958, −5.84326795985066445472506305497, −5.42097680084432441260985158352, −5.11052286252910781006752188929, −5.09344736840694322924578550474, −5.06743989227308102475928794293, −4.53520158093747941873648615536, −4.32658761438328176832168425598, −4.00192230775584855483218061956, −3.83676849180674104860882468450, −3.66681686779937131581053867195, −3.62808881470386944022456487043, −3.04049885620667910171302041284, −2.99202561121724629734560840101, −2.94735370267689146060279006037, −2.69509354155937443838020311932, −2.22164109880770970383911429454, −1.78249951729130500300361076187, −1.62900587843445184857379240011, −1.31830063175629385646288734923, −1.11812944141921607095980171627, −0.51816506731283528899889646883,
0.51816506731283528899889646883, 1.11812944141921607095980171627, 1.31830063175629385646288734923, 1.62900587843445184857379240011, 1.78249951729130500300361076187, 2.22164109880770970383911429454, 2.69509354155937443838020311932, 2.94735370267689146060279006037, 2.99202561121724629734560840101, 3.04049885620667910171302041284, 3.62808881470386944022456487043, 3.66681686779937131581053867195, 3.83676849180674104860882468450, 4.00192230775584855483218061956, 4.32658761438328176832168425598, 4.53520158093747941873648615536, 5.06743989227308102475928794293, 5.09344736840694322924578550474, 5.11052286252910781006752188929, 5.42097680084432441260985158352, 5.84326795985066445472506305497, 5.94605151962651728062059924958, 5.97057968165574326904486221539, 6.21124507125463421976822979471, 6.30781155296319202107101683037
