L(s) = 1 | + 3-s − 2·7-s + 9-s + 4·11-s + 2·13-s + 3·17-s − 2·21-s + 23-s + 27-s + 6·29-s + 7·31-s + 4·33-s + 2·37-s + 2·39-s + 5·41-s − 43-s − 7·47-s − 3·49-s + 3·51-s − 5·53-s − 5·59-s + 8·61-s − 2·63-s − 9·67-s + 69-s + 4·71-s + 8·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.727·17-s − 0.436·21-s + 0.208·23-s + 0.192·27-s + 1.11·29-s + 1.25·31-s + 0.696·33-s + 0.328·37-s + 0.320·39-s + 0.780·41-s − 0.152·43-s − 1.02·47-s − 3/7·49-s + 0.420·51-s − 0.686·53-s − 0.650·59-s + 1.02·61-s − 0.251·63-s − 1.09·67-s + 0.120·69-s + 0.474·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.240824436\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.240824436\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−12.99947396806708, −12.66287522804997, −12.05883853144735, −11.76529242571667, −11.19543780626642, −10.63369407432313, −10.05322879697812, −9.703197053187479, −9.331844976552369, −8.772129906356082, −8.337572760196714, −7.892254432641443, −7.298816297400997, −6.693162528996130, −6.265550670332891, −6.094634416904968, −5.132144726369105, −4.653869343958671, −4.033027516789466, −3.576671992475077, −3.047877877832047, −2.644468261252055, −1.738198946746074, −1.183223147523139, −0.6090190420607420,
0.6090190420607420, 1.183223147523139, 1.738198946746074, 2.644468261252055, 3.047877877832047, 3.576671992475077, 4.033027516789466, 4.653869343958671, 5.132144726369105, 6.094634416904968, 6.265550670332891, 6.693162528996130, 7.298816297400997, 7.892254432641443, 8.337572760196714, 8.772129906356082, 9.331844976552369, 9.703197053187479, 10.05322879697812, 10.63369407432313, 11.19543780626642, 11.76529242571667, 12.05883853144735, 12.66287522804997, 12.99947396806708