L(s) = 1 | + (−1.27 + 0.413i)2-s + (1.09 + 1.09i)3-s + (0.639 − 0.464i)4-s + (−1.85 − 0.944i)6-s + (0.164 − 0.226i)8-s + 1.41i·9-s + (1.21 + 0.192i)12-s + (−0.325 + 0.638i)13-s + (−0.360 + 1.10i)16-s + (−0.585 − 1.80i)18-s + (0.309 + 0.951i)23-s + (0.429 − 0.0680i)24-s + (−0.309 + 0.951i)25-s + (0.150 − 0.947i)26-s + (−0.457 + 0.457i)27-s + ⋯ |
L(s) = 1 | + (−1.27 + 0.413i)2-s + (1.09 + 1.09i)3-s + (0.639 − 0.464i)4-s + (−1.85 − 0.944i)6-s + (0.164 − 0.226i)8-s + 1.41i·9-s + (1.21 + 0.192i)12-s + (−0.325 + 0.638i)13-s + (−0.360 + 1.10i)16-s + (−0.585 − 1.80i)18-s + (0.309 + 0.951i)23-s + (0.429 − 0.0680i)24-s + (−0.309 + 0.951i)25-s + (0.150 − 0.947i)26-s + (−0.457 + 0.457i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7250202520\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7250202520\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.669 + 0.743i)T \) |
good | 2 | \( 1 + (1.27 - 0.413i)T + (0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (-1.09 - 1.09i)T + iT^{2} \) |
| 5 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 11 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 13 | \( 1 + (0.325 - 0.638i)T + (-0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (1.24 + 0.196i)T + (0.951 + 0.309i)T^{2} \) |
| 31 | \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (0.970 + 0.494i)T + (0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 71 | \( 1 + (0.0163 + 0.103i)T + (-0.951 + 0.309i)T^{2} \) |
| 73 | \( 1 + 0.813iT - T^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 97 | \( 1 + (0.951 + 0.309i)T^{2} \) |
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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
−10.08855692253400491694128282716, −9.425135383110892426527583083258, −9.066420909680423845277469526297, −8.226908582434092367795711024339, −7.53942026361073205951675021125, −6.64851291175929894073077799241, −5.20572720753149043257091143722, −4.14083576118655955027482463183, −3.28110629415995091733490812647, −1.84231592711039182908169170396,
1.00765547252638300073012316760, 2.26550828896855368535095418521, 2.89000792570922135211314014994, 4.49065968049740818050439830864, 6.00630687896542402950272865885, 7.07533574877586009598034822496, 7.86947313193422329317731078208, 8.228452966697589253511175456805, 9.064446893067474349427227144408, 9.766411779110085824439079780862