| L(s) = 1 | + 3·2-s + 4-s − 7·7-s − 21·8-s − 60·11-s − 38·13-s − 21·14-s − 71·16-s − 84·17-s + 110·19-s − 180·22-s + 120·23-s − 114·26-s − 7·28-s − 162·29-s + 236·31-s − 45·32-s − 252·34-s + 376·37-s + 330·38-s + 126·41-s + 34·43-s − 60·44-s + 360·46-s − 6·47-s + 49·49-s − 38·52-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s − 0.377·7-s − 0.928·8-s − 1.64·11-s − 0.810·13-s − 0.400·14-s − 1.10·16-s − 1.19·17-s + 1.32·19-s − 1.74·22-s + 1.08·23-s − 0.859·26-s − 0.0472·28-s − 1.03·29-s + 1.36·31-s − 0.248·32-s − 1.27·34-s + 1.67·37-s + 1.40·38-s + 0.479·41-s + 0.120·43-s − 0.205·44-s + 1.15·46-s − 0.0186·47-s + 1/7·49-s − 0.101·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.928835359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.928835359\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 110 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 162 T + p^{3} T^{2} \) |
| 31 | \( 1 - 236 T + p^{3} T^{2} \) |
| 37 | \( 1 - 376 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 34 T + p^{3} T^{2} \) |
| 47 | \( 1 + 6 T + p^{3} T^{2} \) |
| 53 | \( 1 - 582 T + p^{3} T^{2} \) |
| 59 | \( 1 + 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 880 T + p^{3} T^{2} \) |
| 67 | \( 1 - 826 T + p^{3} T^{2} \) |
| 71 | \( 1 - 666 T + p^{3} T^{2} \) |
| 73 | \( 1 - 826 T + p^{3} T^{2} \) |
| 79 | \( 1 + 592 T + p^{3} T^{2} \) |
| 83 | \( 1 - 792 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1002 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1442 T + p^{3} T^{2} \) |
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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
−9.259959779287942174099844048513, −8.132558795758992777576515028532, −7.36225468466772436877315896022, −6.45816995567323247866690154964, −5.48243842081989305299947680214, −4.98086223486882136009533591607, −4.15829616124200138837504344896, −2.93658474986300598401168905161, −2.52040462921003961474678432094, −0.54158829247218443186497927606,
0.54158829247218443186497927606, 2.52040462921003961474678432094, 2.93658474986300598401168905161, 4.15829616124200138837504344896, 4.98086223486882136009533591607, 5.48243842081989305299947680214, 6.45816995567323247866690154964, 7.36225468466772436877315896022, 8.132558795758992777576515028532, 9.259959779287942174099844048513
