L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − i·7-s − i·8-s + 2·9-s + i·12-s + i·13-s + 14-s + 16-s − 3i·17-s + 2i·18-s − 19-s − 21-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.377i·7-s − 0.353i·8-s + 0.666·9-s + 0.288i·12-s + 0.277i·13-s + 0.267·14-s + 0.250·16-s − 0.727i·17-s + 0.471i·18-s − 0.229·19-s − 0.218·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45948 - 0.344538i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45948 - 0.344538i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.871140753453987620504090159995, −9.022994393690980526882549759416, −8.112494231222844401996753182294, −7.27299756019718214529961900449, −6.79808793499429898924222457416, −5.84379151744724813970441112931, −4.71600772933333300382195326786, −3.90831902467415936018583668795, −2.34385773632888722896866027426, −0.803118169931028016795268971038,
1.37013021504630416699966885410, 2.71880443125033685463285670275, 3.80258792603032100861754452958, 4.60404974447699729840986976376, 5.55267944731353413432100877937, 6.62009187741297926155647777340, 7.80253861516502004887216369859, 8.632754664683448521621364142105, 9.499938893504928702570356864974, 10.14412570831628558915944559515