| L(s) = 1 | + 2.56·3-s + 3.56·9-s + 2.12·11-s + 2·13-s − 2.56·17-s + 0.561·19-s − 5.56·23-s + 1.43·27-s + 7.56·29-s + 0.876·31-s + 5.43·33-s + 11.8·37-s + 5.12·39-s + 6.56·41-s − 2.43·43-s + 8.24·47-s − 6.56·51-s − 7.12·53-s + 1.43·57-s + 13.3·59-s − 2.87·61-s − 16.1·67-s − 14.2·69-s − 10.6·71-s + 10.5·73-s + 6.68·79-s − 6.99·81-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 1.18·9-s + 0.640·11-s + 0.554·13-s − 0.621·17-s + 0.128·19-s − 1.15·23-s + 0.276·27-s + 1.40·29-s + 0.157·31-s + 0.946·33-s + 1.94·37-s + 0.820·39-s + 1.02·41-s − 0.371·43-s + 1.20·47-s − 0.918·51-s − 0.978·53-s + 0.190·57-s + 1.74·59-s − 0.368·61-s − 1.96·67-s − 1.71·69-s − 1.26·71-s + 1.23·73-s + 0.752·79-s − 0.777·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.114011803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.114011803\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 - 0.876T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 - 6.56T + 41T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 2.87T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.86106034830499363535280905924, −7.16047932246406090232407885502, −6.32980250793905034625729403157, −5.83259435824048837652576465824, −4.48669435388522843264942160301, −4.18350765372730375175673674396, −3.32600136283598619836183926162, −2.63425221541114374765796023429, −1.93592214692765153906388088179, −0.919684243844775661589864311299,
0.919684243844775661589864311299, 1.93592214692765153906388088179, 2.63425221541114374765796023429, 3.32600136283598619836183926162, 4.18350765372730375175673674396, 4.48669435388522843264942160301, 5.83259435824048837652576465824, 6.32980250793905034625729403157, 7.16047932246406090232407885502, 7.86106034830499363535280905924
