| L(s) = 1 | − 0.0864·7-s + 0.913·11-s − 2.62·13-s + 2.08·17-s + 4.93·19-s + 8.47·23-s − 2.39·29-s + 3.62·31-s + 5.85·37-s − 6.64·41-s + 8.24·43-s + 2.68·47-s − 6.99·49-s − 5.73·53-s − 12.3·59-s − 6.33·61-s − 6.16·67-s + 12.3·71-s − 5.31·73-s − 0.0790·77-s − 13.4·79-s − 6.07·83-s + 8.13·89-s + 0.227·91-s − 11.1·97-s + 12.0·101-s + 4.36·103-s + ⋯ |
| L(s) = 1 | − 0.0326·7-s + 0.275·11-s − 0.728·13-s + 0.506·17-s + 1.13·19-s + 1.76·23-s − 0.445·29-s + 0.651·31-s + 0.962·37-s − 1.03·41-s + 1.25·43-s + 0.392·47-s − 0.998·49-s − 0.787·53-s − 1.60·59-s − 0.811·61-s − 0.753·67-s + 1.47·71-s − 0.621·73-s − 0.00900·77-s − 1.51·79-s − 0.667·83-s + 0.862·89-s + 0.0237·91-s − 1.12·97-s + 1.20·101-s + 0.430·103-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
| \(L(1)\) |
\(\approx\) |
\(2.133098008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.133098008\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0864T + 7T^{2} \) |
| 11 | \( 1 - 0.913T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 + 2.39T + 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 - 5.85T + 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 - 2.68T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 + 6.16T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 5.31T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 6.07T + 83T^{2} \) |
| 89 | \( 1 - 8.13T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.54580557371092028214262405795, −7.37782321060093973566451848944, −6.43971142317183443777008772465, −5.74679461488481796906352453999, −4.95234059939308579058750962286, −4.45517220675694073757228466947, −3.26720472363497753757162357689, −2.89761259198170092083794703100, −1.67625215259235229729200565122, −0.74850251689653631791240457775,
0.74850251689653631791240457775, 1.67625215259235229729200565122, 2.89761259198170092083794703100, 3.26720472363497753757162357689, 4.45517220675694073757228466947, 4.95234059939308579058750962286, 5.74679461488481796906352453999, 6.43971142317183443777008772465, 7.37782321060093973566451848944, 7.54580557371092028214262405795
