| L(s) = 1 | + 28·7-s − 100·25-s + 490·49-s − 376·79-s + 412·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s − 2.80e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 4·7-s − 4·25-s + 10·49-s − 4.75·79-s + 3.40·121-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 − 4·169-s + 0.00578·173-s − 16·175-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 + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.941518321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.941518321\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 206 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 734 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1234 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{2}( 1 + 38 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2}( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 5582 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 118 T + p^{2} T^{2} )^{2}( 1 + 118 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 2914 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
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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
−6.03182199840029981041238123677, −6.01435644029954739315398590299, −5.73838336461493715927733398598, −5.71696302476435179774490964368, −5.58944497305785042878465324992, −5.18498024611397580417224612186, −5.02951750807067246479235640882, −4.85491525035872158458735039785, −4.43587654805635724727378169649, −4.43015272458717487298315479284, −4.29514049478462017125760949066, −3.97844358667334805071987694972, −3.91875061680400934787696279475, −3.53316984955402010779359207033, −3.18831508348986476933961785173, −2.89678992468496095017557884493, −2.53907928507099807550036113178, −2.20658916230527632758580228386, −2.05548956948997599501735072643, −1.81580473106899126449267559015, −1.65553249653308658614203038389, −1.42320477600664260287310872429, −1.07893677092182261150955762017, −0.65562730313945438162156100359, −0.25767565513511899725795939321,
0.25767565513511899725795939321, 0.65562730313945438162156100359, 1.07893677092182261150955762017, 1.42320477600664260287310872429, 1.65553249653308658614203038389, 1.81580473106899126449267559015, 2.05548956948997599501735072643, 2.20658916230527632758580228386, 2.53907928507099807550036113178, 2.89678992468496095017557884493, 3.18831508348986476933961785173, 3.53316984955402010779359207033, 3.91875061680400934787696279475, 3.97844358667334805071987694972, 4.29514049478462017125760949066, 4.43015272458717487298315479284, 4.43587654805635724727378169649, 4.85491525035872158458735039785, 5.02951750807067246479235640882, 5.18498024611397580417224612186, 5.58944497305785042878465324992, 5.71696302476435179774490964368, 5.73838336461493715927733398598, 6.01435644029954739315398590299, 6.03182199840029981041238123677
