L(s) = 1 | + 6·3-s + 21·9-s + 8·13-s + 25-s + 56·27-s + 16·31-s + 16·37-s + 48·39-s − 4·41-s + 18·49-s − 24·53-s − 16·71-s + 6·75-s − 16·79-s + 126·81-s − 16·83-s − 20·89-s + 96·93-s − 24·107-s + 96·111-s + 168·117-s + 34·121-s − 24·123-s − 8·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 3.46·3-s + 7·9-s + 2.21·13-s + 1/5·25-s + 10.7·27-s + 2.87·31-s + 2.63·37-s + 7.68·39-s − 0.624·41-s + 18/7·49-s − 3.29·53-s − 1.89·71-s + 0.692·75-s − 1.80·79-s + 14·81-s − 1.75·83-s − 2.11·89-s + 9.95·93-s − 2.32·107-s + 9.11·111-s + 15.5·117-s + 3.09·121-s − 2.16·123-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(22.04276786\) |
\(L(\frac12)\) |
\(\approx\) |
\(22.04276786\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{6} \) |
| 5 | \( 1 - T^{2} + 8 T^{3} - p T^{4} + p^{3} T^{6} \) |
good | 7 | \( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 191 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 - 4 T + 23 T^{2} - 48 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 2255 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 2 p T^{2} + 1775 T^{4} - 40932 T^{6} + 1775 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 8 T + 89 T^{2} - 432 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 8 T + 3 p T^{2} - 584 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 + 2 T + 23 T^{2} + 220 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 65 T^{2} + 64 T^{3} + 65 p T^{4} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 22367 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 12 T + 191 T^{2} + 1264 T^{3} + 191 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 137 T^{2} - 64 T^{3} + 137 p T^{4} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 133 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 2367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 8 T + 233 T^{2} + 1200 T^{3} + 233 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 8 T + 185 T^{2} + 880 T^{3} + 185 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 10 T + 103 T^{2} + 396 T^{3} + 103 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 39183 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.14578459515129275696764332179, −5.63694126729536772182284383308, −5.63473714897123606512716389006, −5.52535646806503124228940188114, −5.16615964760967576757167229688, −4.81775274407267987981300904447, −4.79601001560153968163757467580, −4.35763830609775652878216371326, −4.33486721493403965482021699712, −4.18008633837503964623073510671, −4.10196148534610072979071011562, −3.85337945841723048975086837887, −3.84981345762096773235078596995, −3.25702569882354700098196248490, −3.06800677854131571684795347737, −3.05054885240179095965044996540, −2.91987301100410838484136628756, −2.72094406423078136074997532728, −2.63028465266689355028116089045, −2.22276113665244492470973079918, −1.78018408387523595314253450485, −1.61786609385062618659483279694, −1.54317322712221154025089766567, −0.962776006244728625407431260328, −0.919617039958997677208070014175,
0.919617039958997677208070014175, 0.962776006244728625407431260328, 1.54317322712221154025089766567, 1.61786609385062618659483279694, 1.78018408387523595314253450485, 2.22276113665244492470973079918, 2.63028465266689355028116089045, 2.72094406423078136074997532728, 2.91987301100410838484136628756, 3.05054885240179095965044996540, 3.06800677854131571684795347737, 3.25702569882354700098196248490, 3.84981345762096773235078596995, 3.85337945841723048975086837887, 4.10196148534610072979071011562, 4.18008633837503964623073510671, 4.33486721493403965482021699712, 4.35763830609775652878216371326, 4.79601001560153968163757467580, 4.81775274407267987981300904447, 5.16615964760967576757167229688, 5.52535646806503124228940188114, 5.63473714897123606512716389006, 5.63694126729536772182284383308, 6.14578459515129275696764332179
Plot not available for L-functions of degree greater than 10.