L(s) = 1 | − 3-s + 9-s + 2·11-s − 2·13-s − 4·17-s + 8·23-s − 27-s − 2·31-s − 2·33-s + 8·37-s + 2·39-s + 2·41-s + 2·43-s − 10·47-s + 4·51-s − 2·53-s − 4·59-s − 10·61-s − 2·67-s − 8·69-s + 12·71-s + 10·73-s − 16·79-s + 81-s + 16·83-s − 14·89-s + 2·93-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.970·17-s + 1.66·23-s − 0.192·27-s − 0.359·31-s − 0.348·33-s + 1.31·37-s + 0.320·39-s + 0.312·41-s + 0.304·43-s − 1.45·47-s + 0.560·51-s − 0.274·53-s − 0.520·59-s − 1.28·61-s − 0.244·67-s − 0.963·69-s + 1.42·71-s + 1.17·73-s − 1.80·79-s + 1/9·81-s + 1.75·83-s − 1.48·89-s + 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.16256979560643, −12.60223923111472, −12.30577818081780, −11.61959074303553, −11.26829438308090, −10.94367730245709, −10.52022472206954, −9.708584512082795, −9.524054231642834, −8.994292594486510, −8.560757958609402, −7.786824313900745, −7.509451973311216, −6.721797759069944, −6.633818315105028, −6.028100765531776, −5.402949843087116, −4.864712870606050, −4.527207162771662, −3.970945599727420, −3.251640623447076, −2.737826859896236, −2.061472297585932, −1.395555060366562, −0.7667401732532654, 0,
0.7667401732532654, 1.395555060366562, 2.061472297585932, 2.737826859896236, 3.251640623447076, 3.970945599727420, 4.527207162771662, 4.864712870606050, 5.402949843087116, 6.028100765531776, 6.633818315105028, 6.721797759069944, 7.509451973311216, 7.786824313900745, 8.560757958609402, 8.994292594486510, 9.524054231642834, 9.708584512082795, 10.52022472206954, 10.94367730245709, 11.26829438308090, 11.61959074303553, 12.30577818081780, 12.60223923111472, 13.16256979560643