L(s) = 1 | + 0.470·2-s + 3-s − 1.77·4-s + 1.57·5-s + 0.470·6-s + 0.714·7-s − 1.77·8-s + 9-s + 0.741·10-s + 11-s − 1.77·12-s + 0.336·14-s + 1.57·15-s + 2.71·16-s + 4.42·17-s + 0.470·18-s + 2.49·19-s − 2.80·20-s + 0.714·21-s + 0.470·22-s − 3.27·23-s − 1.77·24-s − 2.52·25-s + 27-s − 1.27·28-s − 2.28·29-s + 0.741·30-s + ⋯ |
L(s) = 1 | + 0.332·2-s + 0.577·3-s − 0.889·4-s + 0.704·5-s + 0.192·6-s + 0.269·7-s − 0.628·8-s + 0.333·9-s + 0.234·10-s + 0.301·11-s − 0.513·12-s + 0.0898·14-s + 0.406·15-s + 0.679·16-s + 1.07·17-s + 0.110·18-s + 0.572·19-s − 0.626·20-s + 0.155·21-s + 0.100·22-s − 0.682·23-s − 0.363·24-s − 0.504·25-s + 0.192·27-s − 0.240·28-s − 0.424·29-s + 0.135·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.922326977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.922326977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 - 0.714T + 7T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 + 3.27T + 23T^{2} \) |
| 29 | \( 1 + 2.28T + 29T^{2} \) |
| 31 | \( 1 - 3.25T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 6.18T + 41T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 6.76T + 61T^{2} \) |
| 67 | \( 1 - 0.716T + 67T^{2} \) |
| 71 | \( 1 - 6.53T + 71T^{2} \) |
| 73 | \( 1 - 7.63T + 73T^{2} \) |
| 79 | \( 1 - 1.68T + 79T^{2} \) |
| 83 | \( 1 - 5.69T + 83T^{2} \) |
| 89 | \( 1 + 3.00T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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
−8.053517136319275779097511248699, −7.71427150204544860069084623280, −6.62907402502828062936514556702, −5.73390456292583363479054273562, −5.36759869997304924831545240596, −4.37232905503573236646191873158, −3.75212512305825475612566247519, −2.94343554721600851911661411925, −1.92250122011821911792299781169, −0.885688141446869392617900965346,
0.885688141446869392617900965346, 1.92250122011821911792299781169, 2.94343554721600851911661411925, 3.75212512305825475612566247519, 4.37232905503573236646191873158, 5.36759869997304924831545240596, 5.73390456292583363479054273562, 6.62907402502828062936514556702, 7.71427150204544860069084623280, 8.053517136319275779097511248699