| L(s) = 1 | − 2-s + 1.64·3-s + 4-s + 0.930·5-s − 1.64·6-s + 7-s − 8-s − 0.280·9-s − 0.930·10-s − 4.41·11-s + 1.64·12-s + 3.34·13-s − 14-s + 1.53·15-s + 16-s + 1.14·17-s + 0.280·18-s − 6.72·19-s + 0.930·20-s + 1.64·21-s + 4.41·22-s + 8.59·23-s − 1.64·24-s − 4.13·25-s − 3.34·26-s − 5.40·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.952·3-s + 0.5·4-s + 0.416·5-s − 0.673·6-s + 0.377·7-s − 0.353·8-s − 0.0933·9-s − 0.294·10-s − 1.33·11-s + 0.476·12-s + 0.926·13-s − 0.267·14-s + 0.396·15-s + 0.250·16-s + 0.277·17-s + 0.0660·18-s − 1.54·19-s + 0.208·20-s + 0.359·21-s + 0.941·22-s + 1.79·23-s − 0.336·24-s − 0.826·25-s − 0.655·26-s − 1.04·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.093602736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.093602736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 1.64T + 3T^{2} \) |
| 5 | \( 1 - 0.930T + 5T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 + 6.72T + 19T^{2} \) |
| 23 | \( 1 - 8.59T + 23T^{2} \) |
| 29 | \( 1 - 5.75T + 29T^{2} \) |
| 31 | \( 1 - 9.27T + 31T^{2} \) |
| 37 | \( 1 + 5.57T + 37T^{2} \) |
| 41 | \( 1 - 7.96T + 41T^{2} \) |
| 43 | \( 1 - 3.96T + 43T^{2} \) |
| 47 | \( 1 + 2.15T + 47T^{2} \) |
| 53 | \( 1 + 3.87T + 53T^{2} \) |
| 59 | \( 1 + 0.164T + 59T^{2} \) |
| 61 | \( 1 - 4.35T + 61T^{2} \) |
| 67 | \( 1 - 8.90T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 1.79T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 + 3.39T + 97T^{2} \) |
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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
−8.212667855615046449920191947381, −7.74205312910316653846414561103, −6.75991644689857987577612350456, −6.08336477867373178341592083358, −5.28192862461546283467539629207, −4.39723978468411465138290419638, −3.29404404243953215036229045775, −2.62194087922975236715811071390, −2.00122428158378784897578303196, −0.797809921314124288373575609033,
0.797809921314124288373575609033, 2.00122428158378784897578303196, 2.62194087922975236715811071390, 3.29404404243953215036229045775, 4.39723978468411465138290419638, 5.28192862461546283467539629207, 6.08336477867373178341592083358, 6.75991644689857987577612350456, 7.74205312910316653846414561103, 8.212667855615046449920191947381
