L(s) = 1 | − i·3-s + (2.23 − 0.162i)5-s − 0.770i·7-s − 9-s − 0.920·11-s − 3.41i·13-s + (−0.162 − 2.23i)15-s − 2.39i·17-s − 5.29·19-s − 0.770·21-s − 3.81i·23-s + (4.94 − 0.725i)25-s + i·27-s − 3.65·29-s − 10.5·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.997 − 0.0727i)5-s − 0.291i·7-s − 0.333·9-s − 0.277·11-s − 0.947i·13-s + (−0.0420 − 0.575i)15-s − 0.579i·17-s − 1.21·19-s − 0.168·21-s − 0.794i·23-s + (0.989 − 0.145i)25-s + 0.192i·27-s − 0.678·29-s − 1.90·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.012113800\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.012113800\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.23 + 0.162i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 + 0.770iT - 7T^{2} \) |
| 11 | \( 1 + 0.920T + 11T^{2} \) |
| 13 | \( 1 + 3.41iT - 13T^{2} \) |
| 17 | \( 1 + 2.39iT - 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 3.81iT - 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 0.493iT - 37T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 + 4.33iT - 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 2.20T + 59T^{2} \) |
| 61 | \( 1 - 1.58T + 61T^{2} \) |
| 71 | \( 1 + 3.39T + 71T^{2} \) |
| 73 | \( 1 + 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 0.553T + 79T^{2} \) |
| 83 | \( 1 + 7.19iT - 83T^{2} \) |
| 89 | \( 1 + 2.49T + 89T^{2} \) |
| 97 | \( 1 + 9.99iT - 97T^{2} \) |
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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
−8.037422315442803243266735946265, −7.32470002731610392265767869553, −6.60805350381780219032401299274, −5.87101677640391375755605523317, −5.28932947445927361545511181004, −4.37679138552702288799971689702, −3.19211930818630957880910296271, −2.38383215183536282160329521192, −1.52488397123073299837502154410, −0.25741419141062363363904790640,
1.73167351248020382391816274290, 2.27529583091893708012520295004, 3.51555383654871142937881225035, 4.19434850262298846730127057680, 5.29822897635047315667109727742, 5.63795006296362856227528124454, 6.56728702044008037637014748946, 7.17803619565334941320926048947, 8.316958761371476383530161929184, 8.944276284442354764118282742206