L(s) = 1 | + 1.49·2-s − 2.01·3-s + 0.232·4-s − 3.08·5-s − 3.00·6-s − 0.0503·7-s − 2.64·8-s + 1.05·9-s − 4.60·10-s − 4.66·11-s − 0.468·12-s − 13-s − 0.0752·14-s + 6.20·15-s − 4.41·16-s + 5.64·17-s + 1.57·18-s + 5.52·19-s − 0.717·20-s + 0.101·21-s − 6.96·22-s + 2.02·23-s + 5.31·24-s + 4.51·25-s − 1.49·26-s + 3.92·27-s − 0.0117·28-s + ⋯ |
L(s) = 1 | + 1.05·2-s − 1.16·3-s + 0.116·4-s − 1.37·5-s − 1.22·6-s − 0.0190·7-s − 0.933·8-s + 0.350·9-s − 1.45·10-s − 1.40·11-s − 0.135·12-s − 0.277·13-s − 0.0201·14-s + 1.60·15-s − 1.10·16-s + 1.36·17-s + 0.370·18-s + 1.26·19-s − 0.160·20-s + 0.0221·21-s − 1.48·22-s + 0.421·23-s + 1.08·24-s + 0.903·25-s − 0.293·26-s + 0.755·27-s − 0.00221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7804068728\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7804068728\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 3 | \( 1 + 2.01T + 3T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 7 | \( 1 + 0.0503T + 7T^{2} \) |
| 11 | \( 1 + 4.66T + 11T^{2} \) |
| 17 | \( 1 - 5.64T + 17T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 + 9.44T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.15T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 - 7.43T + 53T^{2} \) |
| 59 | \( 1 - 0.277T + 59T^{2} \) |
| 61 | \( 1 - 4.51T + 61T^{2} \) |
| 67 | \( 1 + 6.35T + 67T^{2} \) |
| 71 | \( 1 - 9.56T + 71T^{2} \) |
| 73 | \( 1 + 3.64T + 73T^{2} \) |
| 79 | \( 1 + 5.26T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 - 4.93T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−9.890562980245863013015694284211, −8.628999731974595563275661706573, −7.77910607134996806033051006136, −7.08531747492431444140902488242, −5.94438774725017171998348802812, −5.12466801103037359398936106953, −4.90241527884392994425049869282, −3.60241988816864320225130935910, −2.98042780817761507896586246409, −0.56143307416944863111060308652,
0.56143307416944863111060308652, 2.98042780817761507896586246409, 3.60241988816864320225130935910, 4.90241527884392994425049869282, 5.12466801103037359398936106953, 5.94438774725017171998348802812, 7.08531747492431444140902488242, 7.77910607134996806033051006136, 8.628999731974595563275661706573, 9.890562980245863013015694284211