| L(s) = 1 | − 3·4-s + 6·16-s + 6·19-s − 10·64-s − 18·76-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 3·4-s + 6·16-s + 6·19-s − 10·64-s − 18·76-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.351748572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351748572\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 - T )^{6} \) |
| good | 3 | \( 1 - T^{6} + T^{12} \) |
| 7 | \( 1 - T^{6} + T^{12} \) |
| 11 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 13 | \( 1 - T^{6} + T^{12} \) |
| 17 | \( 1 - T^{6} + T^{12} \) |
| 23 | \( 1 - T^{6} + T^{12} \) |
| 29 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 31 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 41 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 43 | \( ( 1 + T^{2} )^{6} \) |
| 47 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 53 | \( 1 - T^{6} + T^{12} \) |
| 59 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 61 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 67 | \( 1 - T^{6} + T^{12} \) |
| 71 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 73 | \( 1 - T^{6} + T^{12} \) |
| 79 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 83 | \( ( 1 + T^{2} )^{6} \) |
| 89 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 97 | \( ( 1 + T^{2} )^{6} \) |
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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
−4.65903369123427239873299343629, −4.48751680989397356492960479059, −4.28286734584059575301441782150, −4.18243304143084629845907053413, −4.08534998682137289074036942399, −3.72895568218617306746470104714, −3.70882788999299068455384358513, −3.64202305237588747427592660510, −3.38639486701005533871349356420, −3.26243855383392873988199181001, −3.25072671516760740151367110473, −3.11801017304828286099154749620, −3.01108835207740366678947273417, −2.76756033836572864689460323169, −2.57143076711172609993740843198, −2.46947627421263933287602154168, −1.97178740335629982540347773442, −1.90531263182782517530788088820, −1.79346847256458088176360183646, −1.30167715124531853673714111128, −1.29207898681043629964343576588, −1.10111542941000778459311509088, −1.01377433896048908569933917576, −0.62021649994100439373962566133, −0.57320194094828230167610926871,
0.57320194094828230167610926871, 0.62021649994100439373962566133, 1.01377433896048908569933917576, 1.10111542941000778459311509088, 1.29207898681043629964343576588, 1.30167715124531853673714111128, 1.79346847256458088176360183646, 1.90531263182782517530788088820, 1.97178740335629982540347773442, 2.46947627421263933287602154168, 2.57143076711172609993740843198, 2.76756033836572864689460323169, 3.01108835207740366678947273417, 3.11801017304828286099154749620, 3.25072671516760740151367110473, 3.26243855383392873988199181001, 3.38639486701005533871349356420, 3.64202305237588747427592660510, 3.70882788999299068455384358513, 3.72895568218617306746470104714, 4.08534998682137289074036942399, 4.18243304143084629845907053413, 4.28286734584059575301441782150, 4.48751680989397356492960479059, 4.65903369123427239873299343629
Plot not available for L-functions of degree greater than 10.
