| L(s) = 1 | − 3-s + 3·7-s + 9-s + 2·11-s − 13-s + 4·17-s − 3·21-s − 4·23-s − 5·25-s − 27-s − 7·31-s − 2·33-s + 37-s + 39-s − 4·41-s + 43-s − 2·47-s + 2·49-s − 4·51-s − 12·53-s + 2·59-s − 61-s + 3·63-s − 13·67-s + 4·69-s − 10·71-s − 3·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.970·17-s − 0.654·21-s − 0.834·23-s − 25-s − 0.192·27-s − 1.25·31-s − 0.348·33-s + 0.164·37-s + 0.160·39-s − 0.624·41-s + 0.152·43-s − 0.291·47-s + 2/7·49-s − 0.560·51-s − 1.64·53-s + 0.260·59-s − 0.128·61-s + 0.377·63-s − 1.58·67-s + 0.481·69-s − 1.18·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.55344538604637258816744025192, −6.73502980094343579195042138089, −5.88088491098401297690165620228, −5.45673530229714629950464513063, −4.62674360523230339830749344975, −4.04433119163911442361126350256, −3.15542834922238433851016990421, −1.88096236037880357225419052848, −1.39083433986808492707659380632, 0,
1.39083433986808492707659380632, 1.88096236037880357225419052848, 3.15542834922238433851016990421, 4.04433119163911442361126350256, 4.62674360523230339830749344975, 5.45673530229714629950464513063, 5.88088491098401297690165620228, 6.73502980094343579195042138089, 7.55344538604637258816744025192
