| L(s) = 1 | + 2-s + 3-s + 4-s − 3.71·5-s + 6-s + 2.42·7-s + 8-s + 9-s − 3.71·10-s + 5.28·11-s + 12-s + 1.77·13-s + 2.42·14-s − 3.71·15-s + 16-s + 2.42·17-s + 18-s − 2.42·19-s − 3.71·20-s + 2.42·21-s + 5.28·22-s − 23-s + 24-s + 8.76·25-s + 1.77·26-s + 27-s + 2.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.65·5-s + 0.408·6-s + 0.916·7-s + 0.353·8-s + 0.333·9-s − 1.17·10-s + 1.59·11-s + 0.288·12-s + 0.492·13-s + 0.647·14-s − 0.958·15-s + 0.250·16-s + 0.588·17-s + 0.235·18-s − 0.556·19-s − 0.829·20-s + 0.529·21-s + 1.12·22-s − 0.208·23-s + 0.204·24-s + 1.75·25-s + 0.348·26-s + 0.192·27-s + 0.458·28-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
| \(L(1)\) |
\(\approx\) |
\(3.586166020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.586166020\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.71T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 2.42T + 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 31 | \( 1 + 7.19T + 31T^{2} \) |
| 37 | \( 1 + 8.82T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 4.45T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 8.75T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 8.37T + 61T^{2} \) |
| 67 | \( 1 - 5.55T + 67T^{2} \) |
| 71 | \( 1 + 1.49T + 71T^{2} \) |
| 73 | \( 1 + 4.66T + 73T^{2} \) |
| 79 | \( 1 - 6.41T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 + 4.75T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.510052115793032859100934031598, −7.60022360754062212537696992463, −7.16264790735319393075443957970, −6.32642032177584553983700886712, −5.25748536513184171561118047583, −4.35280701315043342251846956946, −3.84923644699699218054776427774, −3.41425895700352252283425616021, −2.05540283793924472189019933892, −1.01791186376631129631626723294,
1.01791186376631129631626723294, 2.05540283793924472189019933892, 3.41425895700352252283425616021, 3.84923644699699218054776427774, 4.35280701315043342251846956946, 5.25748536513184171561118047583, 6.32642032177584553983700886712, 7.16264790735319393075443957970, 7.60022360754062212537696992463, 8.510052115793032859100934031598
