| L(s) = 1 | + 3·7-s + 4·11-s − 7·13-s − 4·17-s + 19-s − 8·23-s + 3·31-s + 2·37-s + 6·41-s + 11·43-s + 6·47-s + 2·49-s − 6·53-s − 6·59-s − 61-s + 15·67-s + 6·71-s + 2·73-s + 12·77-s + 8·79-s + 2·83-s + 16·89-s − 21·91-s + 13·97-s + 2·101-s + 16·103-s − 18·107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 1.20·11-s − 1.94·13-s − 0.970·17-s + 0.229·19-s − 1.66·23-s + 0.538·31-s + 0.328·37-s + 0.937·41-s + 1.67·43-s + 0.875·47-s + 2/7·49-s − 0.824·53-s − 0.781·59-s − 0.128·61-s + 1.83·67-s + 0.712·71-s + 0.234·73-s + 1.36·77-s + 0.900·79-s + 0.219·83-s + 1.69·89-s − 2.20·91-s + 1.31·97-s + 0.199·101-s + 1.57·103-s − 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.114609081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.114609081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.76453929401436297398822473592, −7.42891767330104316799596271663, −6.50985305124455823895936625286, −5.88632367158894428763325829148, −4.84839347282931395897791518872, −4.51061402812677504098904219732, −3.73746159887568839423087272887, −2.40050712533289421428044562097, −1.99359177679202356710728789481, −0.73087120727226290835113748898,
0.73087120727226290835113748898, 1.99359177679202356710728789481, 2.40050712533289421428044562097, 3.73746159887568839423087272887, 4.51061402812677504098904219732, 4.84839347282931395897791518872, 5.88632367158894428763325829148, 6.50985305124455823895936625286, 7.42891767330104316799596271663, 7.76453929401436297398822473592
