L(s) = 1 | − 2-s − 1.63·3-s + 4-s − 2.84·5-s + 1.63·6-s − 8-s − 0.328·9-s + 2.84·10-s + 5.83·11-s − 1.63·12-s + 2.22·13-s + 4.65·15-s + 16-s + 1.51·17-s + 0.328·18-s + 1.82·19-s − 2.84·20-s − 5.83·22-s − 1.53·23-s + 1.63·24-s + 3.10·25-s − 2.22·26-s + 5.44·27-s + 3.43·29-s − 4.65·30-s − 7.63·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.943·3-s + 0.5·4-s − 1.27·5-s + 0.667·6-s − 0.353·8-s − 0.109·9-s + 0.900·10-s + 1.76·11-s − 0.471·12-s + 0.616·13-s + 1.20·15-s + 0.250·16-s + 0.368·17-s + 0.0774·18-s + 0.418·19-s − 0.636·20-s − 1.24·22-s − 0.319·23-s + 0.333·24-s + 0.620·25-s − 0.436·26-s + 1.04·27-s + 0.637·29-s − 0.849·30-s − 1.37·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6833069716\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6833069716\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + 1.63T + 3T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 11 | \( 1 - 5.83T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 + 0.382T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 + 8.71T + 89T^{2} \) |
| 97 | \( 1 - 1.15T + 97T^{2} \) |
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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
−8.612873440868585848349833658005, −7.59361813904495610183572590288, −7.13899775301127120443148204807, −6.20583183987359145438688475279, −5.80572587186550554776281544054, −4.56608210299385663424737799084, −3.87452391278917192512553832077, −3.08234856697033809709450643040, −1.48222195037356589258050387430, −0.59714402035663802350516614070,
0.59714402035663802350516614070, 1.48222195037356589258050387430, 3.08234856697033809709450643040, 3.87452391278917192512553832077, 4.56608210299385663424737799084, 5.80572587186550554776281544054, 6.20583183987359145438688475279, 7.13899775301127120443148204807, 7.59361813904495610183572590288, 8.612873440868585848349833658005