L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.707 − 0.707i)3-s + (0.999 + 1.73i)4-s + (−0.366 − 1.36i)6-s + (3.15 − 3.15i)7-s + 2.82i·8-s + 1.00i·9-s + 2i·11-s + (0.517 − 1.93i)12-s + (1.60 − 1.60i)13-s + (6.09 − 1.63i)14-s + (−2.00 + 3.46i)16-s + (4.24 + 4.24i)17-s + (−0.707 + 1.22i)18-s − 3.19·19-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)2-s + (−0.408 − 0.408i)3-s + (0.499 + 0.866i)4-s + (−0.149 − 0.557i)6-s + (1.19 − 1.19i)7-s + 0.999i·8-s + 0.333i·9-s + 0.603i·11-s + (0.149 − 0.557i)12-s + (0.444 − 0.444i)13-s + (1.62 − 0.436i)14-s + (−0.500 + 0.866i)16-s + (1.02 + 1.02i)17-s + (−0.166 + 0.288i)18-s − 0.733·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.99178 + 0.395936i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99178 + 0.395936i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-3.15 + 3.15i)T - 7iT^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + (-1.60 + 1.60i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 + (5.27 + 5.27i)T + 23iT^{2} \) |
| 29 | \( 1 - 0.535iT - 29T^{2} \) |
| 31 | \( 1 - 3.73iT - 31T^{2} \) |
| 37 | \( 1 + (7.72 + 7.72i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 + (3.15 + 3.15i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.656 - 0.656i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.86 - 3.86i)T - 53iT^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 + (3.91 - 3.91i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.53iT - 71T^{2} \) |
| 73 | \( 1 + (2.82 - 2.82i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 + (-9.14 - 9.14i)T + 83iT^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (6.88 + 6.88i)T + 97iT^{2} \) |
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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
−12.14721725438499034749750919622, −10.83312100424646090270584759830, −10.48726391184732527156937809005, −8.390964352357426250324189037184, −7.78816741810436733350711274342, −6.87780449636749298470432089906, −5.81060862703871976984392050129, −4.70433408026730913379935274600, −3.80052576868758004327729803635, −1.78048616440873034354294552684,
1.77039408276890328389779016259, 3.29619556753170535485728675153, 4.66002522645508902310423901210, 5.47697921565076670252241917958, 6.25962119665459232926629672585, 7.86776973209365021565430803567, 9.038758930999932869463083503129, 10.03223086862074773755999617905, 11.16342007135298985103059804815, 11.69435245978603814277003251284