| L(s) = 1 | − 1.61·2-s − 1.61·3-s + 1.61·4-s + 2.61·6-s + 0.618·7-s − 8-s + 1.61·9-s + 11-s − 2.61·12-s + 13-s − 1.00·14-s − 2.61·18-s − 1.61·19-s − 1.00·21-s − 1.61·22-s + 0.618·23-s + 1.61·24-s + 25-s − 1.61·26-s − 27-s + 1.00·28-s + 32-s − 1.61·33-s + 2.61·36-s + 2.61·38-s − 1.61·39-s + 0.618·41-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 1.61·3-s + 1.61·4-s + 2.61·6-s + 0.618·7-s − 8-s + 1.61·9-s + 11-s − 2.61·12-s + 13-s − 1.00·14-s − 2.61·18-s − 1.61·19-s − 1.00·21-s − 1.61·22-s + 0.618·23-s + 1.61·24-s + 25-s − 1.61·26-s − 27-s + 1.00·28-s + 32-s − 1.61·33-s + 2.61·36-s + 2.61·38-s − 1.61·39-s + 0.618·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2259201697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2259201697\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.61T + T^{2} \) |
| 3 | \( 1 + 1.61T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.618T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−12.93016617607326862101493801299, −11.78795778641297585343072035099, −10.98535866823063653794737106348, −10.62023232874391771421269454926, −9.246236909771999955566743720444, −8.299653780349074161218825853600, −6.88356534604153249087277566380, −6.17153685945518048212164616467, −4.55082127921418290124319818993, −1.36290338861363308272370371699,
1.36290338861363308272370371699, 4.55082127921418290124319818993, 6.17153685945518048212164616467, 6.88356534604153249087277566380, 8.299653780349074161218825853600, 9.246236909771999955566743720444, 10.62023232874391771421269454926, 10.98535866823063653794737106348, 11.78795778641297585343072035099, 12.93016617607326862101493801299
