| L(s) = 1 | − 1.10·3-s − 2.48·5-s − 7-s − 1.76·9-s + 11-s − 13-s + 2.75·15-s − 6.03·17-s + 0.989·19-s + 1.10·21-s + 8.81·23-s + 1.16·25-s + 5.29·27-s − 4.44·29-s + 2.66·31-s − 1.10·33-s + 2.48·35-s − 10.0·37-s + 1.10·39-s − 1.71·41-s − 12.1·43-s + 4.38·45-s − 2.96·47-s + 49-s + 6.69·51-s − 7.19·53-s − 2.48·55-s + ⋯ |
| L(s) = 1 | − 0.640·3-s − 1.10·5-s − 0.377·7-s − 0.589·9-s + 0.301·11-s − 0.277·13-s + 0.711·15-s − 1.46·17-s + 0.226·19-s + 0.242·21-s + 1.83·23-s + 0.232·25-s + 1.01·27-s − 0.826·29-s + 0.479·31-s − 0.193·33-s + 0.419·35-s − 1.65·37-s + 0.177·39-s − 0.268·41-s − 1.85·43-s + 0.654·45-s − 0.433·47-s + 0.142·49-s + 0.937·51-s − 0.987·53-s − 0.334·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3082284464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3082284464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.10T + 3T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 17 | \( 1 + 6.03T + 17T^{2} \) |
| 19 | \( 1 - 0.989T + 19T^{2} \) |
| 23 | \( 1 - 8.81T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 - 2.66T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 1.71T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 + 7.19T + 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 + 0.918T + 61T^{2} \) |
| 67 | \( 1 - 4.50T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 5.38T + 73T^{2} \) |
| 79 | \( 1 + 5.36T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 - 3.26T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.77712247619597559472407326967, −6.87808523335556815780624166363, −6.73071569128102060984230764452, −5.72364437792663385358954340168, −4.94592901789584451386385322008, −4.44439869230457945602410906323, −3.42298508291439579794687011473, −2.93855753711743076370788275495, −1.63673732219487163474573030837, −0.27926487709821426737903592466,
0.27926487709821426737903592466, 1.63673732219487163474573030837, 2.93855753711743076370788275495, 3.42298508291439579794687011473, 4.44439869230457945602410906323, 4.94592901789584451386385322008, 5.72364437792663385358954340168, 6.73071569128102060984230764452, 6.87808523335556815780624166363, 7.77712247619597559472407326967
