| L(s) = 1 | + 4·7-s − 3·9-s − 12·17-s − 8·23-s + 12·31-s − 20·41-s + 8·47-s + 2·49-s − 12·63-s + 8·71-s + 36·73-s − 36·79-s + 6·81-s − 28·89-s − 36·97-s − 4·103-s − 52·113-s − 48·119-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 36·153-s + 157-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 9-s − 2.91·17-s − 1.66·23-s + 2.15·31-s − 3.12·41-s + 1.16·47-s + 2/7·49-s − 1.51·63-s + 0.949·71-s + 4.21·73-s − 4.05·79-s + 2/3·81-s − 2.96·89-s − 3.65·97-s − 0.394·103-s − 4.89·113-s − 4.40·119-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.91·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07990977640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07990977640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 2 T + 5 T^{2} - 12 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 22 T^{2} + 407 T^{4} - 7284 T^{6} + 407 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 6 T + 35 T^{2} + 172 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 4 T + 57 T^{2} + 168 T^{3} + 57 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( ( 1 - 6 T + 77 T^{2} - 308 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 86 T^{2} + 6055 T^{4} - 248372 T^{6} + 6055 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 114 T^{2} + 8087 T^{4} - 416540 T^{6} + 8087 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 4 T + 49 T^{2} + 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{3} \) |
| 59 | \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{3} \) |
| 71 | \( ( 1 - 4 T + 101 T^{2} - 632 T^{3} + 101 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 79 | \( ( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 317 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 274 T^{2} + 37415 T^{4} - 3513756 T^{6} + 37415 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 14 T + 263 T^{2} + 2308 T^{3} + 263 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 + 18 T + 287 T^{2} + 3164 T^{3} + 287 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.58987481826807309364550118405, −4.51174429739290071814014434850, −4.33944013175279623640133187751, −4.24527281148814458843820607764, −4.08559552049577346898908031652, −4.07344265595133827019981963258, −3.86819182125354105459837546483, −3.60934023076752490759228719116, −3.35548540186479242585236426842, −3.35365829398997773389515184498, −2.95486725235835732514757401708, −2.94652229054856263313718280826, −2.76823630522570820480674966336, −2.53740524689382711968890640842, −2.43240132475351963525276057041, −2.21771419985665381855141359655, −2.13118871086503861692011518589, −1.86508079173314118026685808326, −1.67661110172250008711991612569, −1.67124799903151365503596667209, −1.07728734759227355995901152384, −1.07204925848412328350684457097, −1.06293926203455231960508260983, −0.18035925907624018540457071447, −0.07746228627237077411056797527,
0.07746228627237077411056797527, 0.18035925907624018540457071447, 1.06293926203455231960508260983, 1.07204925848412328350684457097, 1.07728734759227355995901152384, 1.67124799903151365503596667209, 1.67661110172250008711991612569, 1.86508079173314118026685808326, 2.13118871086503861692011518589, 2.21771419985665381855141359655, 2.43240132475351963525276057041, 2.53740524689382711968890640842, 2.76823630522570820480674966336, 2.94652229054856263313718280826, 2.95486725235835732514757401708, 3.35365829398997773389515184498, 3.35548540186479242585236426842, 3.60934023076752490759228719116, 3.86819182125354105459837546483, 4.07344265595133827019981963258, 4.08559552049577346898908031652, 4.24527281148814458843820607764, 4.33944013175279623640133187751, 4.51174429739290071814014434850, 4.58987481826807309364550118405
Plot not available for L-functions of degree greater than 10.
