| L(s) = 1 | + 2-s + 0.457·3-s + 4-s + 3.67·5-s + 0.457·6-s − 2.61·7-s + 8-s − 2.79·9-s + 3.67·10-s − 2.87·11-s + 0.457·12-s + 1.06·13-s − 2.61·14-s + 1.67·15-s + 16-s − 4.18·17-s − 2.79·18-s − 19-s + 3.67·20-s − 1.19·21-s − 2.87·22-s + 4.00·23-s + 0.457·24-s + 8.47·25-s + 1.06·26-s − 2.64·27-s − 2.61·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.264·3-s + 0.5·4-s + 1.64·5-s + 0.186·6-s − 0.988·7-s + 0.353·8-s − 0.930·9-s + 1.16·10-s − 0.866·11-s + 0.132·12-s + 0.294·13-s − 0.699·14-s + 0.433·15-s + 0.250·16-s − 1.01·17-s − 0.657·18-s − 0.229·19-s + 0.820·20-s − 0.261·21-s − 0.612·22-s + 0.834·23-s + 0.0933·24-s + 1.69·25-s + 0.208·26-s − 0.509·27-s − 0.494·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.457T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 + 4.18T + 17T^{2} \) |
| 23 | \( 1 - 4.00T + 23T^{2} \) |
| 29 | \( 1 + 0.0627T + 29T^{2} \) |
| 31 | \( 1 + 9.56T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 0.865T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 + 2.99T + 53T^{2} \) |
| 59 | \( 1 + 7.02T + 59T^{2} \) |
| 61 | \( 1 - 7.33T + 61T^{2} \) |
| 67 | \( 1 - 8.24T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.54T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 7.60T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 5.57T + 97T^{2} \) |
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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
−7.13579583561086063341251009917, −6.63981377377963018849237299615, −6.01800978448997696962397590984, −5.40013822868525126148677666130, −4.98318060507132106518335888512, −3.70078629344660988920844176328, −3.01615949943114755193821043890, −2.39837259519856458516075849862, −1.69707264371407927299016258949, 0,
1.69707264371407927299016258949, 2.39837259519856458516075849862, 3.01615949943114755193821043890, 3.70078629344660988920844176328, 4.98318060507132106518335888512, 5.40013822868525126148677666130, 6.01800978448997696962397590984, 6.63981377377963018849237299615, 7.13579583561086063341251009917
