| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 4i·7-s + i·8-s − 9-s − 4·11-s − i·12-s − i·13-s + 4·14-s + 16-s + 2i·17-s + i·18-s + 8·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.51i·7-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s − 0.277i·13-s + 1.06·14-s + 0.250·16-s + 0.485i·17-s + 0.235i·18-s + 1.83·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5032222895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5032222895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 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
−9.442171981004983268700820085736, −9.038790846801555916060558402609, −8.136757369000208394554851343572, −7.43420014116533310936099164253, −5.84179170497446123714677248872, −5.48428596130283810463584322681, −4.73078761309973827507890422768, −3.39800540832981113716495179130, −2.84071337882671718308240376812, −1.78246582324540800448947085834,
0.18127882725460498320601294913, 1.43810909890710103094458964664, 3.01041472238020436378805231295, 3.91166842158406657566510446838, 5.05355820903670978484754566046, 5.57286438697686009944859234868, 6.85381693924992966740704765504, 7.25848202339840041031065293564, 7.77567686522491098667636608820, 8.590477954312880837983564920657
