| L(s) = 1 | − 15·8-s + 27·9-s − 93·31-s − 90·47-s + 161·64-s + 30·67-s − 405·72-s + 486·81-s + 363·121-s − 246·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 507·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.87·8-s + 3·9-s − 3·31-s − 1.91·47-s + 2.51·64-s + 0.447·67-s − 5.62·72-s + 6·81-s + 3·121-s − 1.96·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29791 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29791 ^{s/2} \, \Gamma_{\C}(s+1)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9621602414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9621602414\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 31 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 + 15 T^{3} + p^{6} T^{6} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 5 | $D_{6}$ | \( 1 + 246 T^{3} + p^{6} T^{6} \) |
| 7 | $D_{6}$ | \( 1 + 430 T^{3} + p^{6} T^{6} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 19 | $D_{6}$ | \( 1 - 10618 T^{3} + p^{6} T^{6} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 29 | $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 | $D_{6}$ | \( 1 + 60558 T^{3} + p^{6} T^{6} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 47 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{3} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 59 | $D_{6}$ | \( 1 - 136842 T^{3} + p^{6} T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{3} \) |
| 71 | $D_{6}$ | \( 1 - 284178 T^{3} + p^{6} T^{6} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{3}( 1 + p T )^{3} \) |
| 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 | $D_{6}$ | \( 1 - 1807490 T^{3} + p^{6} T^{6} \) |
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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
−15.14012756956488618529647969673, −14.78764100234381729254012756983, −14.57332882175870503705594733286, −13.81679387811514757617535014104, −13.24259968301571057370083647193, −12.89679906733694773002436370268, −12.66557091308678944657336023126, −12.32709830123748623831638736378, −11.91154700257486403052897418123, −11.05737664690698998741699892475, −11.03650665469179773160258904066, −10.08242677323214070013475925037, −9.892558317412207671577879839023, −9.442905988699880556415678977692, −9.061167385695954258374580885558, −8.469553737557101053224507913069, −7.65844876388251916963154008057, −7.26143369203609402753904570035, −6.75705249553714539170057581473, −6.28119168151866824519828147648, −5.48123695636188947637565074060, −4.80855861602909213501152722686, −3.92924440423807003687096439452, −3.41587889882453997469164657778, −1.86731455675159820796397225746,
1.86731455675159820796397225746, 3.41587889882453997469164657778, 3.92924440423807003687096439452, 4.80855861602909213501152722686, 5.48123695636188947637565074060, 6.28119168151866824519828147648, 6.75705249553714539170057581473, 7.26143369203609402753904570035, 7.65844876388251916963154008057, 8.469553737557101053224507913069, 9.061167385695954258374580885558, 9.442905988699880556415678977692, 9.892558317412207671577879839023, 10.08242677323214070013475925037, 11.03650665469179773160258904066, 11.05737664690698998741699892475, 11.91154700257486403052897418123, 12.32709830123748623831638736378, 12.66557091308678944657336023126, 12.89679906733694773002436370268, 13.24259968301571057370083647193, 13.81679387811514757617535014104, 14.57332882175870503705594733286, 14.78764100234381729254012756983, 15.14012756956488618529647969673
