L(s) = 1 | − 2-s + 3-s + 4-s − 1.56·5-s − 6-s − 1.56·7-s − 8-s + 9-s + 1.56·10-s − 3.12·11-s + 12-s − 5.12·13-s + 1.56·14-s − 1.56·15-s + 16-s + 1.56·17-s − 18-s − 2.43·19-s − 1.56·20-s − 1.56·21-s + 3.12·22-s − 23-s − 24-s − 2.56·25-s + 5.12·26-s + 27-s − 1.56·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.698·5-s − 0.408·6-s − 0.590·7-s − 0.353·8-s + 0.333·9-s + 0.493·10-s − 0.941·11-s + 0.288·12-s − 1.42·13-s + 0.417·14-s − 0.403·15-s + 0.250·16-s + 0.378·17-s − 0.235·18-s − 0.559·19-s − 0.349·20-s − 0.340·21-s + 0.665·22-s − 0.208·23-s − 0.204·24-s − 0.512·25-s + 1.00·26-s + 0.192·27-s − 0.295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7773249132\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7773249132\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 5.12T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.285466327485223598447234240387, −7.73868924089458892654219956396, −7.37511352622942505251297679510, −6.46725894097020761869913520621, −5.53719646869951716146250845634, −4.56947818238381296283825854721, −3.68668950089570768579922628219, −2.75458163773775946608975602764, −2.14982576697332691271305945341, −0.51255432642033881364528420241,
0.51255432642033881364528420241, 2.14982576697332691271305945341, 2.75458163773775946608975602764, 3.68668950089570768579922628219, 4.56947818238381296283825854721, 5.53719646869951716146250845634, 6.46725894097020761869913520621, 7.37511352622942505251297679510, 7.73868924089458892654219956396, 8.285466327485223598447234240387