| L(s) = 1 | + 3.23·3-s + 0.763·7-s + 7.47·9-s + 2.47·21-s − 5.70·23-s + 14.4·27-s − 6·29-s − 4.47·41-s + 11.2·43-s + 13.7·47-s − 6.41·49-s − 13.4·61-s + 5.70·63-s − 8.18·67-s − 18.4·69-s + 24.4·81-s − 17.7·83-s − 19.4·87-s + 6·89-s − 18·101-s + 20.1·103-s − 6.29·107-s + 13.4·109-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 0.288·7-s + 2.49·9-s + 0.539·21-s − 1.19·23-s + 2.78·27-s − 1.11·29-s − 0.698·41-s + 1.71·43-s + 1.99·47-s − 0.916·49-s − 1.71·61-s + 0.719·63-s − 0.999·67-s − 2.22·69-s + 2.71·81-s − 1.94·83-s − 2.08·87-s + 0.635·89-s − 1.79·101-s + 1.98·103-s − 0.608·107-s + 1.28·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.981715976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.981715976\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 - 0.763T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 13.7T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8.18T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.00682755962414974756417550010, −9.249558785995916752160956436642, −8.621293987761356855561274600126, −7.75915321252701140688166020057, −7.26816707798868962597348483819, −5.92584847778616959918884557427, −4.47686394719811505501406495279, −3.71031466964137012738153082018, −2.64624472074562449511516543052, −1.67669514643215980846346607477,
1.67669514643215980846346607477, 2.64624472074562449511516543052, 3.71031466964137012738153082018, 4.47686394719811505501406495279, 5.92584847778616959918884557427, 7.26816707798868962597348483819, 7.75915321252701140688166020057, 8.621293987761356855561274600126, 9.249558785995916752160956436642, 10.00682755962414974756417550010
