L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 15-s + 2·17-s − 21-s + 4·23-s + 25-s − 27-s − 2·29-s + 8·31-s + 35-s + 6·37-s − 2·41-s + 4·43-s + 45-s − 8·47-s + 49-s − 2·51-s − 6·53-s − 8·59-s + 6·61-s + 63-s + 4·67-s − 4·69-s + 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.258·15-s + 0.485·17-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.169·35-s + 0.986·37-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 1.04·59-s + 0.768·61-s + 0.125·63-s + 0.488·67-s − 0.481·69-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.793609347\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.793609347\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.22129102005849, −12.99439086695488, −12.45920672146346, −11.86683659277066, −11.50052325044720, −11.03844325786233, −10.56665222157197, −9.983437473271518, −9.702263719685062, −9.052422844757173, −8.608726319786314, −7.851055061636135, −7.653159562024442, −6.897210907559447, −6.369704466761882, −6.032713278295625, −5.401157773864131, −4.782886489365583, −4.626800922469561, −3.722882287983501, −3.150359558213083, −2.496032973560930, −1.814566824669432, −1.137777675385027, −0.5748630408309294,
0.5748630408309294, 1.137777675385027, 1.814566824669432, 2.496032973560930, 3.150359558213083, 3.722882287983501, 4.626800922469561, 4.782886489365583, 5.401157773864131, 6.032713278295625, 6.369704466761882, 6.897210907559447, 7.653159562024442, 7.851055061636135, 8.608726319786314, 9.052422844757173, 9.702263719685062, 9.983437473271518, 10.56665222157197, 11.03844325786233, 11.50052325044720, 11.86683659277066, 12.45920672146346, 12.99439086695488, 13.22129102005849