| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 6·11-s + 12-s + 2·13-s + 2·14-s + 16-s + 18-s − 4·19-s + 2·21-s + 6·22-s + 23-s + 24-s + 2·26-s + 27-s + 2·28-s − 2·29-s − 8·31-s + 32-s + 6·33-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.436·21-s + 1.27·22-s + 0.208·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s + 1.04·33-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.663524863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.663524863\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.735859407267008093191504789913, −7.75384038612143551481825227611, −7.08202084399045703237058714197, −6.29242503250140612700893359343, −5.62450871844060903987364981935, −4.41465627876916860844741898334, −4.08722976902130833928900626155, −3.20238758355698696966097642301, −2.02222614626540141530342457878, −1.29636951813163619958094100143,
1.29636951813163619958094100143, 2.02222614626540141530342457878, 3.20238758355698696966097642301, 4.08722976902130833928900626155, 4.41465627876916860844741898334, 5.62450871844060903987364981935, 6.29242503250140612700893359343, 7.08202084399045703237058714197, 7.75384038612143551481825227611, 8.735859407267008093191504789913
