| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 3·7-s + 8-s + 9-s + 3·10-s − 2.73·11-s + 12-s + 1.26·13-s + 3·14-s + 3·15-s + 16-s − 6.46·17-s + 18-s + 2.46·19-s + 3·20-s + 3·21-s − 2.73·22-s + 23-s + 24-s + 4·25-s + 1.26·26-s + 27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.34·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 0.333·9-s + 0.948·10-s − 0.823·11-s + 0.288·12-s + 0.351·13-s + 0.801·14-s + 0.774·15-s + 0.250·16-s − 1.56·17-s + 0.235·18-s + 0.565·19-s + 0.670·20-s + 0.654·21-s − 0.582·22-s + 0.208·23-s + 0.204·24-s + 0.800·25-s + 0.248·26-s + 0.192·27-s + 0.566·28-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
| \(L(1)\) |
\(\approx\) |
\(5.452558809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.452558809\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 31 | \( 1 - 0.732T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 1.92T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 6.39T + 61T^{2} \) |
| 67 | \( 1 + 9.46T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 + 4.53T + 83T^{2} \) |
| 89 | \( 1 + 5.46T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.605604054040269641245109358858, −7.56621916691075994063872995211, −7.03410575796934842151850213726, −5.98119933393527183218386044972, −5.48031554669020915385769218925, −4.70908447365688150814569212132, −3.98846431518905623848836320299, −2.66309208802909123954997922720, −2.24520048130962089081873197264, −1.33263568038846512018648282754,
1.33263568038846512018648282754, 2.24520048130962089081873197264, 2.66309208802909123954997922720, 3.98846431518905623848836320299, 4.70908447365688150814569212132, 5.48031554669020915385769218925, 5.98119933393527183218386044972, 7.03410575796934842151850213726, 7.56621916691075994063872995211, 8.605604054040269641245109358858
