| L(s) = 1 | + 12·4-s + 27·9-s + 96·16-s + 75·25-s + 324·36-s + 147·49-s + 640·64-s + 486·81-s + 900·100-s + 127-s + 131-s + 137-s + 139-s + 2.59e3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 1.76e3·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 3·4-s + 3·9-s + 6·16-s + 3·25-s + 9·36-s + 3·49-s + 10·64-s + 6·81-s + 9·100-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 18·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 9·196-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(547^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(547^{3}\right)^{s/2} \, \Gamma_{\C}(s+1)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(15.71602336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.71602336\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 547 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 11 | $D_{6}$ | \( 1 - 474 T^{3} + p^{6} T^{6} \) |
| 13 | $D_{6}$ | \( 1 + 4358 T^{3} + p^{6} T^{6} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 19 | $D_{6}$ | \( 1 + 5974 T^{3} + p^{6} T^{6} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 29 | $D_{6}$ | \( 1 - 40026 T^{3} + p^{6} T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 47 | $D_{6}$ | \( 1 + 57102 T^{3} + p^{6} T^{6} \) |
| 53 | $D_{6}$ | \( 1 - 289002 T^{3} + p^{6} T^{6} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 67 | $D_{6}$ | \( 1 + 363382 T^{3} + p^{6} T^{6} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 73 | $D_{6}$ | \( 1 - 462962 T^{3} + p^{6} T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 97 | $D_{6}$ | \( 1 - 1265218 T^{3} + p^{6} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.769727995935848774830967517245, −9.015700584994313250066262733000, −8.765056203744990769122171382724, −8.659337806684879330641757127344, −7.931137464318274405672144419938, −7.56078683797499917861585537432, −7.46596177983242606053779940563, −7.25580857070273797872079632580, −7.13034977720695553500185836240, −6.62741824993304671830727285447, −6.46471886661242904474601187143, −6.22805564204767418393205385271, −5.96280776225978690011102306678, −5.14943031724711583041757012379, −4.92282029190601306714769922749, −4.91444692951681377133302146445, −3.81995204064117789772822227141, −3.78973971150953595697167912247, −3.65596208128490934707296203806, −2.57196279248095170635693413472, −2.48773918926333368846680989935, −2.46100756500082398882206261707, −1.27754808617544536382403373589, −1.26184553004121869175853389375, −1.22312571509403343849852205344,
1.22312571509403343849852205344, 1.26184553004121869175853389375, 1.27754808617544536382403373589, 2.46100756500082398882206261707, 2.48773918926333368846680989935, 2.57196279248095170635693413472, 3.65596208128490934707296203806, 3.78973971150953595697167912247, 3.81995204064117789772822227141, 4.91444692951681377133302146445, 4.92282029190601306714769922749, 5.14943031724711583041757012379, 5.96280776225978690011102306678, 6.22805564204767418393205385271, 6.46471886661242904474601187143, 6.62741824993304671830727285447, 7.13034977720695553500185836240, 7.25580857070273797872079632580, 7.46596177983242606053779940563, 7.56078683797499917861585537432, 7.931137464318274405672144419938, 8.659337806684879330641757127344, 8.765056203744990769122171382724, 9.015700584994313250066262733000, 9.769727995935848774830967517245
