L(s) = 1 | − 2.64·3-s + 4·7-s + 4.00·9-s + 2.64·11-s + 3·17-s − 2.64·19-s − 10.5·21-s + 4·23-s − 2.64·27-s + 4·31-s − 7.00·33-s + 10.5·37-s + 5·41-s + 5.29·43-s + 8·47-s + 9·49-s − 7.93·51-s − 10.5·53-s + 7.00·57-s + 5.29·59-s − 10.5·61-s + 16.0·63-s + 7.93·67-s − 10.5·69-s − 8·71-s − 7·73-s + 10.5·77-s + ⋯ |
L(s) = 1 | − 1.52·3-s + 1.51·7-s + 1.33·9-s + 0.797·11-s + 0.727·17-s − 0.606·19-s − 2.30·21-s + 0.834·23-s − 0.509·27-s + 0.718·31-s − 1.21·33-s + 1.73·37-s + 0.780·41-s + 0.806·43-s + 1.16·47-s + 1.28·49-s − 1.11·51-s − 1.45·53-s + 0.927·57-s + 0.688·59-s − 1.35·61-s + 2.01·63-s + 0.969·67-s − 1.27·69-s − 0.949·71-s − 0.819·73-s + 1.20·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.675321893\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.675321893\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 7.93T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 7.93T + 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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
−7.82540885708837801000846678398, −7.34013943322064753305140088211, −6.35210926626729670023658311684, −5.96633505550951178748933295280, −5.12933596978929796612457885988, −4.60516689960628174529830578747, −4.01185470517252244226755175794, −2.62044834797461920752246236935, −1.42234667082035164942153810927, −0.835412743000651042412083380861,
0.835412743000651042412083380861, 1.42234667082035164942153810927, 2.62044834797461920752246236935, 4.01185470517252244226755175794, 4.60516689960628174529830578747, 5.12933596978929796612457885988, 5.96633505550951178748933295280, 6.35210926626729670023658311684, 7.34013943322064753305140088211, 7.82540885708837801000846678398