| L(s) = 1 | + 2.39·2-s + 3-s + 3.73·4-s + 3.84·5-s + 2.39·6-s − 2.55·7-s + 4.15·8-s + 9-s + 9.21·10-s + 0.235·11-s + 3.73·12-s + 2.44·13-s − 6.11·14-s + 3.84·15-s + 2.48·16-s + 17-s + 2.39·18-s + 1.83·19-s + 14.3·20-s − 2.55·21-s + 0.563·22-s + 6.80·23-s + 4.15·24-s + 9.80·25-s + 5.86·26-s + 27-s − 9.54·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 0.577·3-s + 1.86·4-s + 1.72·5-s + 0.977·6-s − 0.965·7-s + 1.47·8-s + 0.333·9-s + 2.91·10-s + 0.0709·11-s + 1.07·12-s + 0.679·13-s − 1.63·14-s + 0.993·15-s + 0.621·16-s + 0.242·17-s + 0.564·18-s + 0.420·19-s + 3.21·20-s − 0.557·21-s + 0.120·22-s + 1.41·23-s + 0.848·24-s + 1.96·25-s + 1.15·26-s + 0.192·27-s − 1.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.488553341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.488553341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 - 0.235T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 0.0516T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 1.31T + 47T^{2} \) |
| 53 | \( 1 + 3.78T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 0.795T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 4.27T + 73T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 1.77T + 97T^{2} \) |
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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
−8.548023247282513882655338650809, −7.23066889115605674607394539633, −6.65347511127835617244196468287, −6.11703557835532074915929715507, −5.37753538708665564978485629893, −4.88854491265108428409825929019, −3.48844913026272748128205915676, −3.30584585817075707379899359073, −2.29025669655392391859951868552, −1.51592944113259046011961976042,
1.51592944113259046011961976042, 2.29025669655392391859951868552, 3.30584585817075707379899359073, 3.48844913026272748128205915676, 4.88854491265108428409825929019, 5.37753538708665564978485629893, 6.11703557835532074915929715507, 6.65347511127835617244196468287, 7.23066889115605674607394539633, 8.548023247282513882655338650809
