| L(s) = 1 | + 1.10·2-s − 1.33·3-s − 0.776·4-s + 1.90·5-s − 1.47·6-s − 3.07·8-s − 1.22·9-s + 2.10·10-s + 6.49·11-s + 1.03·12-s − 2.54·15-s − 1.84·16-s + 7.14·17-s − 1.35·18-s + 4.93·19-s − 1.48·20-s + 7.18·22-s − 6.24·23-s + 4.09·24-s − 1.36·25-s + 5.62·27-s − 0.505·29-s − 2.81·30-s − 5.66·31-s + 4.10·32-s − 8.65·33-s + 7.90·34-s + ⋯ |
| L(s) = 1 | + 0.781·2-s − 0.768·3-s − 0.388·4-s + 0.853·5-s − 0.601·6-s − 1.08·8-s − 0.408·9-s + 0.667·10-s + 1.95·11-s + 0.298·12-s − 0.656·15-s − 0.460·16-s + 1.73·17-s − 0.319·18-s + 1.13·19-s − 0.331·20-s + 1.53·22-s − 1.30·23-s + 0.834·24-s − 0.272·25-s + 1.08·27-s − 0.0939·29-s − 0.513·30-s − 1.01·31-s + 0.725·32-s − 1.50·33-s + 1.35·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.498882022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.498882022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 - 1.90T + 5T^{2} \) |
| 11 | \( 1 - 6.49T + 11T^{2} \) |
| 17 | \( 1 - 7.14T + 17T^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 + 0.505T + 29T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 - 0.0345T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 - 0.374T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 2.66T + 53T^{2} \) |
| 59 | \( 1 + 1.28T + 59T^{2} \) |
| 61 | \( 1 - 5.66T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 + 4.37T + 73T^{2} \) |
| 79 | \( 1 - 0.870T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−7.67823100889217437669670035804, −6.81779951220018893414805044890, −5.96997991901369708402707932126, −5.76043830083973594260303338603, −5.30129934536525966511675776403, −4.22770812933237488473438351811, −3.70958975158847048807368818453, −2.91376347557383017175725771948, −1.64524046358266870718108333011, −0.76304914562188599355079184370,
0.76304914562188599355079184370, 1.64524046358266870718108333011, 2.91376347557383017175725771948, 3.70958975158847048807368818453, 4.22770812933237488473438351811, 5.30129934536525966511675776403, 5.76043830083973594260303338603, 5.96997991901369708402707932126, 6.81779951220018893414805044890, 7.67823100889217437669670035804
