L(s) = 1 | + (−1.70 + 0.322i)3-s + 5-s + 1.22i·7-s + (2.79 − 1.09i)9-s + 1.70·11-s + 4.75i·13-s + (−1.70 + 0.322i)15-s + 6.23i·17-s + 7.09·19-s + (−0.396 − 2.09i)21-s − 1.34i·23-s + 25-s + (−4.39 + 2.76i)27-s − 2.58i·29-s + 5.45i·31-s + ⋯ |
L(s) = 1 | + (−0.982 + 0.186i)3-s + 0.447·5-s + 0.464i·7-s + (0.930 − 0.366i)9-s + 0.513·11-s + 1.31i·13-s + (−0.439 + 0.0833i)15-s + 1.51i·17-s + 1.62·19-s + (−0.0865 − 0.456i)21-s − 0.280i·23-s + 0.200·25-s + (−0.846 + 0.533i)27-s − 0.480i·29-s + 0.980i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.436123299\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.436123299\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.322i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (0.267 + 8.18i)T \) |
good | 7 | \( 1 - 1.22iT - 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 4.75iT - 13T^{2} \) |
| 17 | \( 1 - 6.23iT - 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 + 1.34iT - 23T^{2} \) |
| 29 | \( 1 + 2.58iT - 29T^{2} \) |
| 31 | \( 1 - 5.45iT - 31T^{2} \) |
| 37 | \( 1 + 6.93T + 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 - 1.89iT - 43T^{2} \) |
| 47 | \( 1 + 0.286iT - 47T^{2} \) |
| 53 | \( 1 + 3.48T + 53T^{2} \) |
| 59 | \( 1 + 0.962iT - 59T^{2} \) |
| 61 | \( 1 - 1.51iT - 61T^{2} \) |
| 71 | \( 1 + 6.96iT - 71T^{2} \) |
| 73 | \( 1 - 0.332T + 73T^{2} \) |
| 79 | \( 1 - 8.41iT - 79T^{2} \) |
| 83 | \( 1 + 9.17iT - 83T^{2} \) |
| 89 | \( 1 + 4.45iT - 89T^{2} \) |
| 97 | \( 1 - 9.67iT - 97T^{2} \) |
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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
−8.922408799474633020864356301438, −7.84556026469553269282025347451, −6.91591103585586741068601130998, −6.39503598859083700637019157013, −5.73491780287326806084968065127, −5.01971275527824129562183731088, −4.20015117518544403997244275452, −3.39155683442798135989240002797, −1.97504419392968637133434658598, −1.22414271922201000278176440100,
0.54815561998006472073611728531, 1.32481136172013319862207098433, 2.68872888049853754529483853556, 3.61618742507027186262863528066, 4.68192464401259906761693183859, 5.44559200718386321580401065343, 5.76990203648172740484468655618, 6.94255812554815118737613760640, 7.26206522147145370966552908402, 8.012947727266431270113223261496