L(s) = 1 | + (−0.707 + 0.707i)2-s + (−2.28 + 2.28i)3-s − 1.00i·4-s − 3.23i·6-s + (0.835 − 2.51i)7-s + (0.707 + 0.707i)8-s − 7.46i·9-s − 1.73·11-s + (2.28 + 2.28i)12-s + (1.67 − 1.67i)13-s + (1.18 + 2.36i)14-s − 1.00·16-s + (−3.96 − 3.96i)17-s + (5.27 + 5.27i)18-s + 5.60·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−1.32 + 1.32i)3-s − 0.500i·4-s − 1.32i·6-s + (0.315 − 0.948i)7-s + (0.250 + 0.250i)8-s − 2.48i·9-s − 0.522·11-s + (0.660 + 0.660i)12-s + (0.464 − 0.464i)13-s + (0.316 + 0.632i)14-s − 0.250·16-s + (−0.960 − 0.960i)17-s + (1.24 + 1.24i)18-s + 1.28·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.553556 + 0.00307106i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.553556 + 0.00307106i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.835 + 2.51i)T \) |
good | 3 | \( 1 + (2.28 - 2.28i)T - 3iT^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + (-1.67 + 1.67i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.96 + 3.96i)T + 17iT^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 + (-1.55 - 1.55i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.26iT - 29T^{2} \) |
| 31 | \( 1 - 4.10iT - 31T^{2} \) |
| 37 | \( 1 + (-5.79 + 5.79i)T - 37iT^{2} \) |
| 41 | \( 1 + 9.70iT - 41T^{2} \) |
| 43 | \( 1 + (2.44 + 2.44i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.90 + 2.90i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.68 - 2.68i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 11.2iT - 61T^{2} \) |
| 67 | \( 1 + (-2.77 + 2.77i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 + (-5.18 + 5.18i)T - 73iT^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + (-6.86 + 6.86i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 + (3.34 + 3.34i)T + 97iT^{2} \) |
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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
−11.04037780777400573579931271135, −10.66703013282457352335250478036, −9.769869131653696758606084247404, −8.994348636573283651240974373846, −7.55534959690094792613541521527, −6.62977467868566855916577841574, −5.41908800991089798559897728933, −4.85130152059748643044506041874, −3.60690258153112071186208516471, −0.59693673742613665601423935523,
1.33394170271769521340681001037, 2.51663416413654839926662124701, 4.72959850320007058533838502127, 5.85199138109300203678683311498, 6.62165334437653331494652307944, 7.77697514971383512454712380932, 8.477141915132868342978840442993, 9.799144250571539259977801182824, 11.05800194095124615747084230100, 11.42749917436028794850910340487