L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 2·10-s − 13-s − 16-s + 17-s + 4·19-s − 2·20-s − 25-s + 26-s − 2·29-s − 8·31-s − 5·32-s − 34-s − 2·37-s − 4·38-s + 6·40-s + 2·41-s + 4·43-s − 8·47-s − 7·49-s + 50-s + 52-s + 10·53-s + 2·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s − 0.277·13-s − 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.447·20-s − 1/5·25-s + 0.196·26-s − 0.371·29-s − 1.43·31-s − 0.883·32-s − 0.171·34-s − 0.328·37-s − 0.648·38-s + 0.948·40-s + 0.312·41-s + 0.609·43-s − 1.16·47-s − 49-s + 0.141·50-s + 0.138·52-s + 1.37·53-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.115755644\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.115755644\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.87568087071730, −12.56461022937170, −11.93781644231470, −11.41507419624735, −10.80890572079546, −10.52889067595910, −9.931015980085062, −9.531584438501555, −9.277964234359986, −8.891408685622776, −8.174779412945481, −7.730715232027278, −7.422851073978474, −6.775382794940181, −6.226255164826374, −5.491148593230041, −5.376750707865814, −4.738229612849764, −4.126499954246640, −3.526332601989912, −2.999529958843753, −2.134279150528041, −1.717274041482087, −1.150636179216224, −0.3494102564665872,
0.3494102564665872, 1.150636179216224, 1.717274041482087, 2.134279150528041, 2.999529958843753, 3.526332601989912, 4.126499954246640, 4.738229612849764, 5.376750707865814, 5.491148593230041, 6.226255164826374, 6.775382794940181, 7.422851073978474, 7.730715232027278, 8.174779412945481, 8.891408685622776, 9.277964234359986, 9.531584438501555, 9.931015980085062, 10.52889067595910, 10.80890572079546, 11.41507419624735, 11.93781644231470, 12.56461022937170, 12.87568087071730