L(s) = 1 | + 3-s − 5-s − 2·9-s − 3·11-s − 13-s − 15-s − 5·17-s + 6·19-s + 25-s − 5·27-s + 5·29-s + 2·31-s − 3·33-s + 4·37-s − 39-s − 2·41-s − 10·43-s + 2·45-s − 9·47-s − 5·51-s − 6·53-s + 3·55-s + 6·57-s + 6·59-s + 12·61-s + 65-s + 2·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s − 0.277·13-s − 0.258·15-s − 1.21·17-s + 1.37·19-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 0.359·31-s − 0.522·33-s + 0.657·37-s − 0.160·39-s − 0.312·41-s − 1.52·43-s + 0.298·45-s − 1.31·47-s − 0.700·51-s − 0.824·53-s + 0.404·55-s + 0.794·57-s + 0.781·59-s + 1.53·61-s + 0.124·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.363121147\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.363121147\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.00666505856340, −15.40391328411178, −14.92591662706138, −14.31389737379146, −13.82460674701587, −13.19362219118236, −12.87718261312465, −11.91552869195590, −11.48010270724289, −11.13752759806509, −10.10398464440020, −9.883006482155254, −9.000387321111059, −8.435379147245880, −8.077141690557460, −7.387991789450450, −6.771048465963531, −6.031653051633737, −5.164366516963955, −4.783477981961983, −3.854292051729348, −3.034283232679340, −2.695008256878966, −1.738127860764370, −0.4692855450168438,
0.4692855450168438, 1.738127860764370, 2.695008256878966, 3.034283232679340, 3.854292051729348, 4.783477981961983, 5.164366516963955, 6.031653051633737, 6.771048465963531, 7.387991789450450, 8.077141690557460, 8.435379147245880, 9.000387321111059, 9.883006482155254, 10.10398464440020, 11.13752759806509, 11.48010270724289, 11.91552869195590, 12.87718261312465, 13.19362219118236, 13.82460674701587, 14.31389737379146, 14.92591662706138, 15.40391328411178, 16.00666505856340