| L(s) = 1 | − 2-s − 3-s + 4-s − 0.891·5-s + 6-s + 4.07·7-s − 8-s + 9-s + 0.891·10-s − 4.41·11-s − 12-s − 13-s − 4.07·14-s + 0.891·15-s + 16-s − 3.83·17-s − 18-s + 1.31·19-s − 0.891·20-s − 4.07·21-s + 4.41·22-s − 6.87·23-s + 24-s − 4.20·25-s + 26-s − 27-s + 4.07·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.398·5-s + 0.408·6-s + 1.54·7-s − 0.353·8-s + 0.333·9-s + 0.281·10-s − 1.32·11-s − 0.288·12-s − 0.277·13-s − 1.09·14-s + 0.230·15-s + 0.250·16-s − 0.929·17-s − 0.235·18-s + 0.302·19-s − 0.199·20-s − 0.889·21-s + 0.940·22-s − 1.43·23-s + 0.204·24-s − 0.841·25-s + 0.196·26-s − 0.192·27-s + 0.770·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5958259533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5958259533\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 0.891T + 5T^{2} \) |
| 7 | \( 1 - 4.07T + 7T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 17 | \( 1 + 3.83T + 17T^{2} \) |
| 19 | \( 1 - 1.31T + 19T^{2} \) |
| 23 | \( 1 + 6.87T + 23T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 + 4.56T + 31T^{2} \) |
| 37 | \( 1 + 8.36T + 37T^{2} \) |
| 41 | \( 1 + 0.436T + 41T^{2} \) |
| 43 | \( 1 + 3.87T + 43T^{2} \) |
| 47 | \( 1 - 1.61T + 47T^{2} \) |
| 53 | \( 1 - 4.54T + 53T^{2} \) |
| 59 | \( 1 - 3.89T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 3.87T + 67T^{2} \) |
| 71 | \( 1 + 8.92T + 71T^{2} \) |
| 73 | \( 1 - 9.37T + 73T^{2} \) |
| 79 | \( 1 - 8.17T + 79T^{2} \) |
| 83 | \( 1 - 4.84T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 1.63T + 97T^{2} \) |
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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
−7.76769491240316815559902844081, −7.46124154811833741490504172525, −6.61075226331877944532566758493, −5.48269564212382258440864408427, −5.31941498242415667996724696644, −4.36446188301668761661289231554, −3.59660132185162473601653427421, −2.12861719972307270700483442979, −1.90569829809568410654147462011, −0.41991649614098024565453622333,
0.41991649614098024565453622333, 1.90569829809568410654147462011, 2.12861719972307270700483442979, 3.59660132185162473601653427421, 4.36446188301668761661289231554, 5.31941498242415667996724696644, 5.48269564212382258440864408427, 6.61075226331877944532566758493, 7.46124154811833741490504172525, 7.76769491240316815559902844081
