L(s) = 1 | − 2-s + 1.99·3-s + 4-s + 0.264·5-s − 1.99·6-s − 4.95·7-s − 8-s + 0.964·9-s − 0.264·10-s − 3.87·11-s + 1.99·12-s − 4.49·13-s + 4.95·14-s + 0.527·15-s + 16-s + 0.946·17-s − 0.964·18-s + 5.96·19-s + 0.264·20-s − 9.87·21-s + 3.87·22-s − 8.40·23-s − 1.99·24-s − 4.92·25-s + 4.49·26-s − 4.05·27-s − 4.95·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.14·3-s + 0.5·4-s + 0.118·5-s − 0.812·6-s − 1.87·7-s − 0.353·8-s + 0.321·9-s − 0.0837·10-s − 1.16·11-s + 0.574·12-s − 1.24·13-s + 1.32·14-s + 0.136·15-s + 0.250·16-s + 0.229·17-s − 0.227·18-s + 1.36·19-s + 0.0591·20-s − 2.15·21-s + 0.826·22-s − 1.75·23-s − 0.406·24-s − 0.985·25-s + 0.882·26-s − 0.779·27-s − 0.936·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8803997536\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8803997536\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
good | 3 | \( 1 - 1.99T + 3T^{2} \) |
| 5 | \( 1 - 0.264T + 5T^{2} \) |
| 7 | \( 1 + 4.95T + 7T^{2} \) |
| 11 | \( 1 + 3.87T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 0.946T + 17T^{2} \) |
| 19 | \( 1 - 5.96T + 19T^{2} \) |
| 23 | \( 1 + 8.40T + 23T^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 37 | \( 1 - 4.50T + 37T^{2} \) |
| 41 | \( 1 - 6.96T + 41T^{2} \) |
| 43 | \( 1 - 8.79T + 43T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 - 8.84T + 53T^{2} \) |
| 59 | \( 1 - 0.388T + 59T^{2} \) |
| 61 | \( 1 - 9.00T + 61T^{2} \) |
| 67 | \( 1 - 6.28T + 67T^{2} \) |
| 71 | \( 1 + 0.478T + 71T^{2} \) |
| 73 | \( 1 - 7.20T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 3.19T + 83T^{2} \) |
| 89 | \( 1 + 3.86T + 89T^{2} \) |
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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
−7.922020329192402595548798348690, −7.59590355708722500844601211760, −7.03338905769528167848577524484, −5.83510212823039376667629172068, −5.62706024269117912888856466549, −4.06698009303592907415402946216, −3.38050429396101702434467363387, −2.53753021995245955696319206862, −2.30893935444836199909779190834, −0.47261484077782584968936667121,
0.47261484077782584968936667121, 2.30893935444836199909779190834, 2.53753021995245955696319206862, 3.38050429396101702434467363387, 4.06698009303592907415402946216, 5.62706024269117912888856466549, 5.83510212823039376667629172068, 7.03338905769528167848577524484, 7.59590355708722500844601211760, 7.922020329192402595548798348690