L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 3·7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 3.81·13-s − 3·14-s + 15-s + 16-s + 17-s − 18-s + 7.81·19-s + 20-s + 3·21-s + 22-s + 5.81·23-s − 24-s + 25-s + 3.81·26-s + 27-s + 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.05·13-s − 0.801·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.79·19-s + 0.223·20-s + 0.654·21-s + 0.213·22-s + 1.21·23-s − 0.204·24-s + 0.200·25-s + 0.748·26-s + 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.785379611\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.785379611\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 3.81T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 23 | \( 1 - 5.81T + 23T^{2} \) |
| 29 | \( 1 + 6.81T + 29T^{2} \) |
| 31 | \( 1 - 6.81T + 31T^{2} \) |
| 41 | \( 1 + 4.81T + 41T^{2} \) |
| 43 | \( 1 + 4.81T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 - 6.81T + 61T^{2} \) |
| 67 | \( 1 - 7.63T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 3.81T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 4.81T + 97T^{2} \) |
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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
−9.773127164759084043386419234376, −9.068457563127379902859084169114, −8.173719141728138354802397577553, −7.55464997660694362544493713208, −6.87115962064833637747421136068, −5.41713309124767003739671321208, −4.86390960009462647397940138963, −3.29472424029316505048279098558, −2.30233594787457595819060582330, −1.21532669289439810366363743417,
1.21532669289439810366363743417, 2.30233594787457595819060582330, 3.29472424029316505048279098558, 4.86390960009462647397940138963, 5.41713309124767003739671321208, 6.87115962064833637747421136068, 7.55464997660694362544493713208, 8.173719141728138354802397577553, 9.068457563127379902859084169114, 9.773127164759084043386419234376