| L(s) = 1 | + 2-s − 1.14·3-s + 4-s + 2.89·5-s − 1.14·6-s + 3.60·7-s + 8-s − 1.68·9-s + 2.89·10-s + 3.08·11-s − 1.14·12-s + 1.07·13-s + 3.60·14-s − 3.32·15-s + 16-s + 2.58·17-s − 1.68·18-s − 19-s + 2.89·20-s − 4.13·21-s + 3.08·22-s − 1.16·23-s − 1.14·24-s + 3.36·25-s + 1.07·26-s + 5.37·27-s + 3.60·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.662·3-s + 0.5·4-s + 1.29·5-s − 0.468·6-s + 1.36·7-s + 0.353·8-s − 0.560·9-s + 0.914·10-s + 0.930·11-s − 0.331·12-s + 0.297·13-s + 0.963·14-s − 0.857·15-s + 0.250·16-s + 0.626·17-s − 0.396·18-s − 0.229·19-s + 0.646·20-s − 0.903·21-s + 0.657·22-s − 0.242·23-s − 0.234·24-s + 0.673·25-s + 0.210·26-s + 1.03·27-s + 0.681·28-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
| \(L(1)\) |
\(\approx\) |
\(4.467320906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.467320906\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.14T + 3T^{2} \) |
| 5 | \( 1 - 2.89T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 3.08T + 11T^{2} \) |
| 13 | \( 1 - 1.07T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 - 5.13T + 29T^{2} \) |
| 31 | \( 1 + 0.677T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 8.91T + 71T^{2} \) |
| 73 | \( 1 - 3.33T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 4.72T + 83T^{2} \) |
| 89 | \( 1 + 0.242T + 89T^{2} \) |
| 97 | \( 1 - 0.265T + 97T^{2} \) |
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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
−7.71738723261984273888068465402, −6.87618715501485424882716241311, −6.11075588354248501832079542119, −5.81185729002915916788536050145, −5.08898764111127411184604828245, −4.57893161150547478981847007041, −3.61104695435015851782348601511, −2.57934108324374437138626236891, −1.76007926848280292273210703853, −1.07030158927293895282753892596,
1.07030158927293895282753892596, 1.76007926848280292273210703853, 2.57934108324374437138626236891, 3.61104695435015851782348601511, 4.57893161150547478981847007041, 5.08898764111127411184604828245, 5.81185729002915916788536050145, 6.11075588354248501832079542119, 6.87618715501485424882716241311, 7.71738723261984273888068465402
