| L(s) = 1 | + 3.43·3-s + 0.919·5-s + 8.78·9-s + 1.60·11-s − 13-s + 3.15·15-s − 2.72·17-s − 2.11·19-s + 5.43·23-s − 4.15·25-s + 19.8·27-s − 6.50·29-s + 5.66·31-s + 5.50·33-s + 9.43·37-s − 3.43·39-s − 2.35·41-s − 0.513·43-s + 8.07·45-s + 4.63·47-s − 9.34·51-s + 5.62·53-s + 1.47·55-s − 7.27·57-s + 7.33·59-s − 0.606·61-s − 0.919·65-s + ⋯ |
| L(s) = 1 | + 1.98·3-s + 0.411·5-s + 2.92·9-s + 0.483·11-s − 0.277·13-s + 0.814·15-s − 0.660·17-s − 0.485·19-s + 1.13·23-s − 0.831·25-s + 3.82·27-s − 1.20·29-s + 1.01·31-s + 0.958·33-s + 1.55·37-s − 0.549·39-s − 0.367·41-s − 0.0783·43-s + 1.20·45-s + 0.675·47-s − 1.30·51-s + 0.773·53-s + 0.198·55-s − 0.963·57-s + 0.955·59-s − 0.0776·61-s − 0.114·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.293267452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.293267452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.43T + 3T^{2} \) |
| 5 | \( 1 - 0.919T + 5T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 17 | \( 1 + 2.72T + 17T^{2} \) |
| 19 | \( 1 + 2.11T + 19T^{2} \) |
| 23 | \( 1 - 5.43T + 23T^{2} \) |
| 29 | \( 1 + 6.50T + 29T^{2} \) |
| 31 | \( 1 - 5.66T + 31T^{2} \) |
| 37 | \( 1 - 9.43T + 37T^{2} \) |
| 41 | \( 1 + 2.35T + 41T^{2} \) |
| 43 | \( 1 + 0.513T + 43T^{2} \) |
| 47 | \( 1 - 4.63T + 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 - 7.33T + 59T^{2} \) |
| 61 | \( 1 + 0.606T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 8.52T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 7.21T + 97T^{2} \) |
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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
−8.855127218950067542807836997437, −8.343842824050729802793666493201, −7.42570931632072399732811061635, −6.95046381976788297798573087568, −5.90760481367996961752671849123, −4.54338179945622296085997921112, −4.02710402466343820849236833735, −2.97811561985514131895200235573, −2.31631927459285674634008565335, −1.38809660610194867148642542191,
1.38809660610194867148642542191, 2.31631927459285674634008565335, 2.97811561985514131895200235573, 4.02710402466343820849236833735, 4.54338179945622296085997921112, 5.90760481367996961752671849123, 6.95046381976788297798573087568, 7.42570931632072399732811061635, 8.343842824050729802793666493201, 8.855127218950067542807836997437
