L(s) = 1 | + 2-s − 3-s + 4-s − 3.84·5-s − 6-s + 8-s + 9-s − 3.84·10-s − 0.480·11-s − 12-s − 2.48·13-s + 3.84·15-s + 16-s + 17-s + 18-s − 1.36·19-s − 3.84·20-s − 0.480·22-s + 2·23-s − 24-s + 9.80·25-s − 2.48·26-s − 27-s − 4.32·29-s + 3.84·30-s − 4.96·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.72·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.21·10-s − 0.144·11-s − 0.288·12-s − 0.687·13-s + 0.993·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.313·19-s − 0.860·20-s − 0.102·22-s + 0.417·23-s − 0.204·24-s + 1.96·25-s − 0.486·26-s − 0.192·27-s − 0.803·29-s + 0.702·30-s − 0.890·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.175775111\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.175775111\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 3.84T + 5T^{2} \) |
| 11 | \( 1 + 0.480T + 11T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 4.32T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 + 8.17T + 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 + 1.84T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 - 0.151T + 53T^{2} \) |
| 59 | \( 1 - 8.32T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 + 0.735T + 71T^{2} \) |
| 73 | \( 1 - 8.80T + 73T^{2} \) |
| 79 | \( 1 - 4.07T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 7.54T + 89T^{2} \) |
| 97 | \( 1 + 1.11T + 97T^{2} \) |
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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
−7.969281193046056584684883728902, −7.37549454521678063504826940806, −6.94398252866536428362704646727, −6.00108046664700285279405389721, −5.07699743091321251832021979975, −4.64184077304474591196631722471, −3.74295399707831946574013061535, −3.26151859002765599982264832747, −1.99563523140195700186009690573, −0.53042401826385715886070170243,
0.53042401826385715886070170243, 1.99563523140195700186009690573, 3.26151859002765599982264832747, 3.74295399707831946574013061535, 4.64184077304474591196631722471, 5.07699743091321251832021979975, 6.00108046664700285279405389721, 6.94398252866536428362704646727, 7.37549454521678063504826940806, 7.969281193046056584684883728902