L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 2·11-s − 12-s + 2·13-s − 15-s + 16-s − 4·17-s + 18-s + 20-s + 2·22-s + 8·23-s − 24-s + 25-s + 2·26-s − 27-s − 30-s − 2·31-s + 32-s − 2·33-s − 4·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.223·20-s + 0.426·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.182·30-s − 0.359·31-s + 0.176·32-s − 0.348·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.551067624\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.551067624\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.438318904212609891056159882541, −8.893883320761320325127642296502, −7.67452669104186096573712814722, −6.71448030263619857726292129732, −6.26866033555376129285213632855, −5.32234060729049876880170623804, −4.56361178578709645761945260896, −3.61784426087409943129297947529, −2.41846334213277678783501330003, −1.14838616509578909491858361321,
1.14838616509578909491858361321, 2.41846334213277678783501330003, 3.61784426087409943129297947529, 4.56361178578709645761945260896, 5.32234060729049876880170623804, 6.26866033555376129285213632855, 6.71448030263619857726292129732, 7.67452669104186096573712814722, 8.893883320761320325127642296502, 9.438318904212609891056159882541