| L(s) = 1 | + 4·3-s − 4·5-s + 4·7-s + 8·9-s + 8·11-s + 6·13-s − 16·15-s − 6·17-s + 16·21-s − 12·23-s + 11·25-s + 12·27-s − 4·29-s + 32·33-s − 16·35-s + 6·37-s + 24·39-s − 12·43-s − 32·45-s + 12·47-s + 8·49-s − 24·51-s − 6·53-s − 32·55-s + 32·63-s − 24·65-s + 12·67-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 1.51·7-s + 8/3·9-s + 2.41·11-s + 1.66·13-s − 4.13·15-s − 1.45·17-s + 3.49·21-s − 2.50·23-s + 11/5·25-s + 2.30·27-s − 0.742·29-s + 5.57·33-s − 2.70·35-s + 0.986·37-s + 3.84·39-s − 1.82·43-s − 4.77·45-s + 1.75·47-s + 8/7·49-s − 3.36·51-s − 0.824·53-s − 4.31·55-s + 4.03·63-s − 2.97·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.408675154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.408675154\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.78000155244330814722909761744, −10.49953090973320678114938046979, −9.595837863586471341676557871521, −9.106066377149385055643637010456, −8.973402191308713849250181184958, −8.507285587563172611323430280318, −8.277194263967601432643402914382, −7.80462613316474300730407221303, −7.76502993630664451722651564132, −6.93457328775907997463927279719, −6.47946745621833698467095916299, −6.11135016692257192349242371734, −4.93993805467126066029941428681, −4.34712349221846685579644102941, −3.97752863634768898288480899700, −3.74578683485653843943465030451, −3.46382502047251490782923274322, −2.37312902068764667396529395229, −1.85686263916154129754932742731, −1.16603695945558309403988358106,
1.16603695945558309403988358106, 1.85686263916154129754932742731, 2.37312902068764667396529395229, 3.46382502047251490782923274322, 3.74578683485653843943465030451, 3.97752863634768898288480899700, 4.34712349221846685579644102941, 4.93993805467126066029941428681, 6.11135016692257192349242371734, 6.47946745621833698467095916299, 6.93457328775907997463927279719, 7.76502993630664451722651564132, 7.80462613316474300730407221303, 8.277194263967601432643402914382, 8.507285587563172611323430280318, 8.973402191308713849250181184958, 9.106066377149385055643637010456, 9.595837863586471341676557871521, 10.49953090973320678114938046979, 10.78000155244330814722909761744
