| L(s) = 1 | − 3·9-s + 12·11-s + 6·19-s + 24·41-s + 18·49-s + 8·59-s + 28·61-s + 32·71-s + 32·79-s + 6·81-s − 36·99-s + 44·101-s + 20·109-s + 66·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 58·169-s − 18·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 9-s + 3.61·11-s + 1.37·19-s + 3.74·41-s + 18/7·49-s + 1.04·59-s + 3.58·61-s + 3.79·71-s + 3.60·79-s + 2/3·81-s − 3.61·99-s + 4.37·101-s + 1.91·109-s + 6·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.46·169-s − 1.37·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(52.20314809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(52.20314809\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( ( 1 - T )^{6} \) |
| good | 7 | \( 1 - 18 T^{2} + 111 T^{4} - 460 T^{6} + 111 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 6 T + 21 T^{2} - 56 T^{3} + 21 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 58 T^{2} + 1607 T^{4} - 26428 T^{6} + 1607 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 10 T^{2} + 303 T^{4} - 9164 T^{6} + 303 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 + 18 T^{2} + 57 p T^{4} + 14652 T^{6} + 57 p^{3} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 63 T^{2} + 36 T^{3} + 63 p T^{4} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 21 T^{2} - 208 T^{3} + 21 p T^{4} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 42 T^{2} - 969 T^{4} + 85412 T^{6} - 969 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 12 T + 147 T^{2} - 988 T^{3} + 147 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 74 T^{2} + 7175 T^{4} - 283388 T^{6} + 7175 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 126 T^{2} + 6543 T^{4} - 260836 T^{6} + 6543 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 82 T^{2} + 9367 T^{4} + 462812 T^{6} + 9367 p^{2} T^{8} + 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 4 T + 33 T^{2} + 392 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 14 T + 227 T^{2} - 1700 T^{3} + 227 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 82 T^{2} + 10247 T^{4} - 648604 T^{6} + 10247 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 16 T + 149 T^{2} - 960 T^{3} + 149 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 234 T^{2} + 29631 T^{4} - 2502604 T^{6} + 29631 p^{2} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 16 T + 3 p T^{2} - 2144 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 454 T^{2} + 89223 T^{4} - 9698804 T^{6} + 89223 p^{2} T^{8} - 454 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 243 T^{2} + 36 T^{3} + 243 p T^{4} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 530 T^{2} + 121295 T^{4} - 15382892 T^{6} + 121295 p^{2} T^{8} - 530 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.30664220152786171565962230030, −3.94240091929509544741526529652, −3.72921039810134132515393790486, −3.68235457144366501519054140382, −3.56287568496010836692297703465, −3.50703047542574954466688367009, −3.50286962593426809745361721518, −3.40272697085866075703622303376, −2.93735105833201849296368313429, −2.84316729182076293004943236226, −2.82982319275267651540892595669, −2.55126941085165675300421880034, −2.46528648645024665072641386789, −2.09674015552643546535285387791, −2.08037912943365585739863573039, −1.99878059473678688126697746687, −1.92179247036934687920652130782, −1.63752596451745472802188754285, −1.52634412193335121002692261512, −0.953886992627638478802563206222, −0.901808010059099602905627644111, −0.76075255572067327470633421339, −0.71132318654055739438942251500, −0.64927631806626242206423101939, −0.62588112750439738957519416075,
0.62588112750439738957519416075, 0.64927631806626242206423101939, 0.71132318654055739438942251500, 0.76075255572067327470633421339, 0.901808010059099602905627644111, 0.953886992627638478802563206222, 1.52634412193335121002692261512, 1.63752596451745472802188754285, 1.92179247036934687920652130782, 1.99878059473678688126697746687, 2.08037912943365585739863573039, 2.09674015552643546535285387791, 2.46528648645024665072641386789, 2.55126941085165675300421880034, 2.82982319275267651540892595669, 2.84316729182076293004943236226, 2.93735105833201849296368313429, 3.40272697085866075703622303376, 3.50286962593426809745361721518, 3.50703047542574954466688367009, 3.56287568496010836692297703465, 3.68235457144366501519054140382, 3.72921039810134132515393790486, 3.94240091929509544741526529652, 4.30664220152786171565962230030
Plot not available for L-functions of degree greater than 10.
