L(s) = 1 | + 0.676·2-s − 3-s − 1.54·4-s − 1.80·5-s − 0.676·6-s − 4.44·7-s − 2.39·8-s + 9-s − 1.22·10-s − 1.20·11-s + 1.54·12-s − 3.05·13-s − 3.00·14-s + 1.80·15-s + 1.46·16-s − 6.57·17-s + 0.676·18-s − 2.24·19-s + 2.78·20-s + 4.44·21-s − 0.814·22-s − 23-s + 2.39·24-s − 1.73·25-s − 2.06·26-s − 27-s + 6.85·28-s + ⋯ |
L(s) = 1 | + 0.478·2-s − 0.577·3-s − 0.771·4-s − 0.807·5-s − 0.276·6-s − 1.68·7-s − 0.847·8-s + 0.333·9-s − 0.386·10-s − 0.362·11-s + 0.445·12-s − 0.847·13-s − 0.803·14-s + 0.466·15-s + 0.365·16-s − 1.59·17-s + 0.159·18-s − 0.514·19-s + 0.622·20-s + 0.970·21-s − 0.173·22-s − 0.208·23-s + 0.489·24-s − 0.347·25-s − 0.405·26-s − 0.192·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1302117296\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1302117296\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 0.676T + 2T^{2} \) |
| 5 | \( 1 + 1.80T + 5T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 1.20T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 + 6.57T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 37 | \( 1 + 8.63T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 + 8.54T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 + 0.352T + 61T^{2} \) |
| 67 | \( 1 - 1.02T + 67T^{2} \) |
| 71 | \( 1 + 3.08T + 71T^{2} \) |
| 73 | \( 1 + 8.06T + 73T^{2} \) |
| 79 | \( 1 - 4.13T + 79T^{2} \) |
| 83 | \( 1 + 1.05T + 83T^{2} \) |
| 89 | \( 1 - 2.79T + 89T^{2} \) |
| 97 | \( 1 - 8.49T + 97T^{2} \) |
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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
−9.169341538801868643250707660369, −8.517232568036867919317184407595, −7.41420399387724238910764075184, −6.64260533106541082858677551156, −6.00092279301179236486645061691, −4.99797255075645579596104146269, −4.25234591866386174172466179503, −3.55279531965561941314505334824, −2.53203189230540012192199565552, −0.21398763684061749363130051731,
0.21398763684061749363130051731, 2.53203189230540012192199565552, 3.55279531965561941314505334824, 4.25234591866386174172466179503, 4.99797255075645579596104146269, 6.00092279301179236486645061691, 6.64260533106541082858677551156, 7.41420399387724238910764075184, 8.517232568036867919317184407595, 9.169341538801868643250707660369