L(s) = 1 | − 1.41·3-s − 1.41·7-s + 1.00·9-s + 1.41·11-s − 17-s + 2.00·21-s + 1.41·23-s + 25-s + 1.41·31-s − 2.00·33-s + 1.00·49-s + 1.41·51-s + 2·53-s − 1.41·63-s − 2.00·69-s − 1.41·71-s − 1.41·75-s − 2.00·77-s + 1.41·79-s − 0.999·81-s − 2.00·93-s + 1.41·99-s − 1.41·107-s + 1.41·119-s + ⋯ |
L(s) = 1 | − 1.41·3-s − 1.41·7-s + 1.00·9-s + 1.41·11-s − 17-s + 2.00·21-s + 1.41·23-s + 25-s + 1.41·31-s − 2.00·33-s + 1.00·49-s + 1.41·51-s + 2·53-s − 1.41·63-s − 2.00·69-s − 1.41·71-s − 1.41·75-s − 2.00·77-s + 1.41·79-s − 0.999·81-s − 2.00·93-s + 1.41·99-s − 1.41·107-s + 1.41·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5779876145\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5779876145\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + 1.41T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 2T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−10.21167418230782988748523401063, −9.296066038871673136654449650049, −8.713650738826188438592156819796, −6.99039660730034060091348377053, −6.66397939773053260846861252611, −6.04260430333767708789146036035, −4.97220783726061437243871088380, −4.05699066288755176613180912150, −2.87056178504683542145951033814, −0.941932617348772231688619127313,
0.941932617348772231688619127313, 2.87056178504683542145951033814, 4.05699066288755176613180912150, 4.97220783726061437243871088380, 6.04260430333767708789146036035, 6.66397939773053260846861252611, 6.99039660730034060091348377053, 8.713650738826188438592156819796, 9.296066038871673136654449650049, 10.21167418230782988748523401063