| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 2.82·7-s − 8-s + 9-s + 10-s + 5.65·11-s + 12-s + 2.82·14-s − 15-s + 16-s − 4.82·17-s − 18-s + 2.82·19-s − 20-s − 2.82·21-s − 5.65·22-s + 8.48·23-s − 24-s + 25-s + 27-s − 2.82·28-s − 3.17·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 1.06·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.70·11-s + 0.288·12-s + 0.755·14-s − 0.258·15-s + 0.250·16-s − 1.17·17-s − 0.235·18-s + 0.648·19-s − 0.223·20-s − 0.617·21-s − 1.20·22-s + 1.76·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.534·28-s − 0.588·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.450007070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450007070\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−8.468545166225628927029892392593, −7.42693948834442370517097038804, −6.83244680036359649470495983994, −6.54926498629110669761379222494, −5.40638072026252363427293184397, −4.25786046620005344885319922450, −3.56463387544722458430995814050, −2.91883604987481559382582709013, −1.78868874857924587396892691233, −0.71910151700348068788922255322,
0.71910151700348068788922255322, 1.78868874857924587396892691233, 2.91883604987481559382582709013, 3.56463387544722458430995814050, 4.25786046620005344885319922450, 5.40638072026252363427293184397, 6.54926498629110669761379222494, 6.83244680036359649470495983994, 7.42693948834442370517097038804, 8.468545166225628927029892392593
