| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s + 13-s + 16-s + 6·17-s + 18-s + 4·23-s + 24-s + 26-s + 27-s − 10·29-s + 32-s + 6·34-s + 36-s + 6·37-s + 39-s + 2·41-s + 4·43-s + 4·46-s + 48-s − 7·49-s + 6·51-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.834·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 1.85·29-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.986·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.589·46-s + 0.144·48-s − 49-s + 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.642554850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.642554850\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.293252484969654106304049481823, −8.234879489422851945430306826139, −7.60700872158310494193298642929, −6.86438211519676053988082818155, −5.82962581845731117670630851923, −5.19858177043401418936672862873, −4.07503419069323835222144097411, −3.40427025338568566249547589245, −2.47214823780223903426583871299, −1.25086079084129802130227088861,
1.25086079084129802130227088861, 2.47214823780223903426583871299, 3.40427025338568566249547589245, 4.07503419069323835222144097411, 5.19858177043401418936672862873, 5.82962581845731117670630851923, 6.86438211519676053988082818155, 7.60700872158310494193298642929, 8.234879489422851945430306826139, 9.293252484969654106304049481823
