L(s) = 1 | − 3-s − 4i·7-s − 2·9-s − 3i·11-s + i·17-s − 7i·19-s + 4i·21-s + 4i·23-s + 5·27-s − 8i·29-s − 4·31-s + 3i·33-s − 4·37-s + 3·41-s + 8·43-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51i·7-s − 0.666·9-s − 0.904i·11-s + 0.242i·17-s − 1.60i·19-s + 0.872i·21-s + 0.834i·23-s + 0.962·27-s − 1.48i·29-s − 0.718·31-s + 0.522i·33-s − 0.657·37-s + 0.468·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5154983854\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5154983854\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 + 9T + 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - T + 83T^{2} \) |
| 89 | \( 1 + 13T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.150857942610757783595936539909, −7.46810545084120630596574467585, −6.72702726690258594056126029637, −6.00375224817096469608045648535, −5.24255833252512168199715169886, −4.34613104888088534349692802905, −3.57703563113304929716118509773, −2.61431931021041846750722624654, −1.05773738251783684196571010817, −0.19358975436772309142960306434,
1.65085841388583397591508596478, 2.54203858623779619495287386620, 3.47664571005465371696842967054, 4.71481752233690790434291693863, 5.36802178796702621293015918707, 5.95945423322715080601183065980, 6.63591557567047728067448045055, 7.64015623870181745785009872076, 8.421765471740855846526678920095, 9.049560697432852885851378854964