L(s) = 1 | + 2-s + 3-s + 4-s + 4.01·5-s + 6-s + 2.23·7-s + 8-s + 9-s + 4.01·10-s − 5.19·11-s + 12-s + 13-s + 2.23·14-s + 4.01·15-s + 16-s + 7.80·17-s + 18-s + 1.23·19-s + 4.01·20-s + 2.23·21-s − 5.19·22-s − 2.23·23-s + 24-s + 11.1·25-s + 26-s + 27-s + 2.23·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.79·5-s + 0.408·6-s + 0.845·7-s + 0.353·8-s + 0.333·9-s + 1.26·10-s − 1.56·11-s + 0.288·12-s + 0.277·13-s + 0.597·14-s + 1.03·15-s + 0.250·16-s + 1.89·17-s + 0.235·18-s + 0.283·19-s + 0.897·20-s + 0.488·21-s − 1.10·22-s − 0.465·23-s + 0.204·24-s + 2.22·25-s + 0.196·26-s + 0.192·27-s + 0.422·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.787940848\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.787940848\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 4.01T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 17 | \( 1 - 7.80T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 + 0.116T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 + 4.93T + 37T^{2} \) |
| 41 | \( 1 - 3.24T + 41T^{2} \) |
| 43 | \( 1 - 9.31T + 43T^{2} \) |
| 47 | \( 1 - 1.50T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 + 1.08T + 61T^{2} \) |
| 67 | \( 1 + 5.69T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 0.0792T + 79T^{2} \) |
| 83 | \( 1 + 9.59T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 3.88T + 97T^{2} \) |
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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
−7.66377171812084768119867015914, −7.31883159776434256739733531950, −6.02440571356840801172787726508, −5.75012302155413169992263958793, −5.14595752475856931079527689683, −4.48713141250674054784751647812, −3.23677727073889117318322690724, −2.73784315737658300870766037836, −1.90140802492242869210670232971, −1.27817084898797039268920571187,
1.27817084898797039268920571187, 1.90140802492242869210670232971, 2.73784315737658300870766037836, 3.23677727073889117318322690724, 4.48713141250674054784751647812, 5.14595752475856931079527689683, 5.75012302155413169992263958793, 6.02440571356840801172787726508, 7.31883159776434256739733531950, 7.66377171812084768119867015914