| L(s) = 1 | + 1.20·3-s − 2.88·5-s − 4.09·7-s − 1.54·9-s − 1.48·11-s − 4.73·13-s − 3.47·15-s − 5.78·17-s − 19-s − 4.93·21-s − 5.93·23-s + 3.32·25-s − 5.47·27-s + 1.35·29-s − 3.88·31-s − 1.78·33-s + 11.8·35-s + 4.39·37-s − 5.70·39-s − 4.60·41-s − 4.40·43-s + 4.47·45-s + 5.66·47-s + 9.77·49-s − 6.96·51-s + 10.0·53-s + 4.28·55-s + ⋯ |
| L(s) = 1 | + 0.695·3-s − 1.29·5-s − 1.54·7-s − 0.516·9-s − 0.447·11-s − 1.31·13-s − 0.897·15-s − 1.40·17-s − 0.229·19-s − 1.07·21-s − 1.23·23-s + 0.664·25-s − 1.05·27-s + 0.251·29-s − 0.698·31-s − 0.311·33-s + 1.99·35-s + 0.723·37-s − 0.913·39-s − 0.719·41-s − 0.671·43-s + 0.666·45-s + 0.826·47-s + 1.39·49-s − 0.975·51-s + 1.37·53-s + 0.577·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03112639163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03112639163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 + 4.09T + 7T^{2} \) |
| 11 | \( 1 + 1.48T + 11T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 - 1.35T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 - 4.39T + 37T^{2} \) |
| 41 | \( 1 + 4.60T + 41T^{2} \) |
| 43 | \( 1 + 4.40T + 43T^{2} \) |
| 47 | \( 1 - 5.66T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 0.570T + 61T^{2} \) |
| 67 | \( 1 - 7.78T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 + 6.88T + 73T^{2} \) |
| 83 | \( 1 - 3.98T + 83T^{2} \) |
| 89 | \( 1 - 0.268T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.033697471456201596736850681675, −7.47710196099756370574828979317, −6.80434539965399646720278168323, −6.10127109245746802074545218499, −5.11821462816471293246656127556, −4.13642156854043808812141293204, −3.66753134948943862927796950536, −2.76875206073751687645844152750, −2.29072851208768048397126485735, −0.080479194630697109028055297397,
0.080479194630697109028055297397, 2.29072851208768048397126485735, 2.76875206073751687645844152750, 3.66753134948943862927796950536, 4.13642156854043808812141293204, 5.11821462816471293246656127556, 6.10127109245746802074545218499, 6.80434539965399646720278168323, 7.47710196099756370574828979317, 8.033697471456201596736850681675
