L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.46 − 0.931i)3-s − 1.00i·4-s + (−1.69 + 0.373i)6-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (1.26 + 2.72i)9-s + 6.30i·11-s + (−0.931 + 1.46i)12-s + (0.977 − 0.977i)13-s − 1.00·14-s − 1.00·16-s + (−4.86 + 4.86i)17-s + (2.81 + 1.02i)18-s + 0.285i·19-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.843 − 0.537i)3-s − 0.500i·4-s + (−0.690 + 0.152i)6-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + (0.421 + 0.906i)9-s + 1.90i·11-s + (−0.268 + 0.421i)12-s + (0.271 − 0.271i)13-s − 0.267·14-s − 0.250·16-s + (−1.18 + 1.18i)17-s + (0.664 + 0.242i)18-s + 0.0655i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.132361313\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.132361313\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.46 + 0.931i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 - 6.30iT - 11T^{2} \) |
| 13 | \( 1 + (-0.977 + 0.977i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.86 - 4.86i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.285iT - 19T^{2} \) |
| 23 | \( 1 + (-4.26 - 4.26i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.84T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 + (0.317 + 0.317i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.55iT - 41T^{2} \) |
| 43 | \( 1 + (2.07 - 2.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.69 - 6.69i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.12 - 3.12i)T + 53iT^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 1.09T + 61T^{2} \) |
| 67 | \( 1 + (-5.63 - 5.63i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.42iT - 71T^{2} \) |
| 73 | \( 1 + (-3.69 + 3.69i)T - 73iT^{2} \) |
| 79 | \( 1 - 4.38iT - 79T^{2} \) |
| 83 | \( 1 + (1.52 + 1.52i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.96T + 89T^{2} \) |
| 97 | \( 1 + (1.50 + 1.50i)T + 97iT^{2} \) |
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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
−10.18044952356847515471744988875, −9.475029282733681523783376709734, −8.196293593657796098213631408600, −7.12358687678681331654900473552, −6.63835539239965973218719673677, −5.61048665471157410904855968926, −4.70737897281991080459215228021, −3.95555841166621494451117432310, −2.36552194559870983222299493498, −1.39757920981732774624858466586,
0.50406427720037678625452491275, 2.81731224341935341468366480756, 3.76755823562504217973082773206, 4.82612413024921342307570729791, 5.52045516698086781357099910325, 6.45905069515973627476261607548, 6.84510772452417792599737347532, 8.366654040926687265667194358354, 8.906519627157106648573547260878, 9.829916473396644224785886934974