L(s) = 1 | − 3-s + 9-s + 4·11-s − 13-s + 2·17-s − 4·19-s + 8·23-s − 27-s − 6·29-s − 4·33-s + 10·37-s + 39-s + 6·41-s − 4·43-s − 7·49-s − 2·51-s − 6·53-s + 4·57-s − 12·59-s − 10·61-s − 12·67-s − 8·69-s + 12·71-s + 10·73-s − 8·79-s + 81-s + 16·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 0.192·27-s − 1.11·29-s − 0.696·33-s + 1.64·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s − 49-s − 0.280·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s − 1.28·61-s − 1.46·67-s − 0.963·69-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 1/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.920547236\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.920547236\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.99150977149807, −14.71812425009435, −14.10756353796041, −13.43398561187305, −12.73506214275075, −12.61201118148764, −11.83921113425685, −11.26411701601260, −11.00960216982605, −10.35925421992983, −9.553036340165049, −9.263541825375780, −8.750843664025031, −7.721230463552079, −7.551409614013287, −6.592033498620420, −6.313532716499060, −5.732943940240804, −4.737357902288262, −4.622957786983533, −3.648618571244595, −3.120620484860122, −2.106551761741310, −1.368777726087921, −0.5848115592201166,
0.5848115592201166, 1.368777726087921, 2.106551761741310, 3.120620484860122, 3.648618571244595, 4.622957786983533, 4.737357902288262, 5.732943940240804, 6.313532716499060, 6.592033498620420, 7.551409614013287, 7.721230463552079, 8.750843664025031, 9.263541825375780, 9.553036340165049, 10.35925421992983, 11.00960216982605, 11.26411701601260, 11.83921113425685, 12.61201118148764, 12.73506214275075, 13.43398561187305, 14.10756353796041, 14.71812425009435, 14.99150977149807