L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 4.44·7-s + 8-s + 9-s + 3.44·11-s + 12-s − 2.44·13-s + 4.44·14-s + 16-s + 4.44·17-s + 18-s + 19-s + 4.44·21-s + 3.44·22-s − 23-s + 24-s − 2.44·26-s + 27-s + 4.44·28-s − 4.34·29-s − 3·31-s + 32-s + 3.44·33-s + 4.44·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.68·7-s + 0.353·8-s + 0.333·9-s + 1.04·11-s + 0.288·12-s − 0.679·13-s + 1.18·14-s + 0.250·16-s + 1.07·17-s + 0.235·18-s + 0.229·19-s + 0.970·21-s + 0.735·22-s − 0.208·23-s + 0.204·24-s − 0.480·26-s + 0.192·27-s + 0.840·28-s − 0.807·29-s − 0.538·31-s + 0.176·32-s + 0.600·33-s + 0.763·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.669738499\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.669738499\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 + 0.898T + 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 + 7.44T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 8.34T + 83T^{2} \) |
| 89 | \( 1 - 2.10T + 89T^{2} \) |
| 97 | \( 1 - 1.55T + 97T^{2} \) |
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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
−8.653749543697073343755866390903, −7.88639056193247196846963749607, −7.41193256573456384568363244441, −6.53279609354272689207936999325, −5.40164789597058649026960225906, −4.92822969354287169868296173550, −4.01342753890289662619039195054, −3.29320357549393105643964805691, −2.02789944767745113755249956991, −1.40369334751528457070149741084,
1.40369334751528457070149741084, 2.02789944767745113755249956991, 3.29320357549393105643964805691, 4.01342753890289662619039195054, 4.92822969354287169868296173550, 5.40164789597058649026960225906, 6.53279609354272689207936999325, 7.41193256573456384568363244441, 7.88639056193247196846963749607, 8.653749543697073343755866390903