L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 4·11-s + 12-s + 2·13-s + 14-s + 16-s − 2·17-s − 18-s − 4·19-s − 21-s − 4·22-s + 8·23-s − 24-s − 2·26-s + 27-s − 28-s + 6·29-s − 8·31-s − 32-s + 4·33-s + 2·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.500296459\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.500296459\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.625809353569461036085862289118, −8.993802635293947884746939912145, −8.601351251693854628821228011897, −7.41750004971397288593425742052, −6.73892644153891083442310873080, −5.95251363997533671735290275834, −4.45776004306556863633827269854, −3.51293692634747222502503411630, −2.39285800255617234808224415611, −1.08656149587708795900736124088,
1.08656149587708795900736124088, 2.39285800255617234808224415611, 3.51293692634747222502503411630, 4.45776004306556863633827269854, 5.95251363997533671735290275834, 6.73892644153891083442310873080, 7.41750004971397288593425742052, 8.601351251693854628821228011897, 8.993802635293947884746939912145, 9.625809353569461036085862289118