| L(s) = 1 | + 7-s + 4·13-s + 6·17-s + 2·19-s + 3·23-s + 3·29-s − 10·31-s + 10·37-s + 9·41-s + 4·43-s − 9·47-s − 6·49-s + 6·53-s − 6·59-s − 61-s − 11·67-s + 12·71-s + 4·73-s − 10·79-s + 9·83-s + 9·89-s + 4·91-s + 10·97-s − 18·101-s − 8·103-s − 9·107-s − 19·109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.10·13-s + 1.45·17-s + 0.458·19-s + 0.625·23-s + 0.557·29-s − 1.79·31-s + 1.64·37-s + 1.40·41-s + 0.609·43-s − 1.31·47-s − 6/7·49-s + 0.824·53-s − 0.781·59-s − 0.128·61-s − 1.34·67-s + 1.42·71-s + 0.468·73-s − 1.12·79-s + 0.987·83-s + 0.953·89-s + 0.419·91-s + 1.01·97-s − 1.79·101-s − 0.788·103-s − 0.870·107-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.620046247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.620046247\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.85183412692991280268209321471, −7.27216683550176494467412774295, −6.34434458519697831088386530205, −5.73985647429667001325056621147, −5.12887363326197211119950499689, −4.22921397807776347434098543844, −3.49051681868049102661843268989, −2.78682666271959125150566575636, −1.59168497822819124815584438671, −0.873023944465869602770010516044,
0.873023944465869602770010516044, 1.59168497822819124815584438671, 2.78682666271959125150566575636, 3.49051681868049102661843268989, 4.22921397807776347434098543844, 5.12887363326197211119950499689, 5.73985647429667001325056621147, 6.34434458519697831088386530205, 7.27216683550176494467412774295, 7.85183412692991280268209321471
