L(s) = 1 | − i·2-s − 1.84i·3-s − 4-s − 3.23i·5-s − 1.84·6-s + i·8-s − 0.414·9-s − 3.23·10-s + (3.28 − 0.468i)11-s + 1.84i·12-s − 6.10·13-s − 5.98·15-s + 16-s − 8.13·17-s + 0.414i·18-s + 6.61·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.06i·3-s − 0.5·4-s − 1.44i·5-s − 0.754·6-s + 0.353i·8-s − 0.138·9-s − 1.02·10-s + (0.989 − 0.141i)11-s + 0.533i·12-s − 1.69·13-s − 1.54·15-s + 0.250·16-s − 1.97·17-s + 0.0976i·18-s + 1.51·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.067513706\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.067513706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.28 + 0.468i)T \) |
good | 3 | \( 1 + 1.84iT - 3T^{2} \) |
| 5 | \( 1 + 3.23iT - 5T^{2} \) |
| 13 | \( 1 + 6.10T + 13T^{2} \) |
| 17 | \( 1 + 8.13T + 17T^{2} \) |
| 19 | \( 1 - 6.61T + 19T^{2} \) |
| 23 | \( 1 + 5.30T + 23T^{2} \) |
| 29 | \( 1 + 1.32iT - 29T^{2} \) |
| 31 | \( 1 + 2.35iT - 31T^{2} \) |
| 37 | \( 1 - 5.28T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 - 3.73iT - 43T^{2} \) |
| 47 | \( 1 + 1.70iT - 47T^{2} \) |
| 53 | \( 1 + 3.98T + 53T^{2} \) |
| 59 | \( 1 - 9.03iT - 59T^{2} \) |
| 61 | \( 1 - 8.55T + 61T^{2} \) |
| 67 | \( 1 - 2.35T + 67T^{2} \) |
| 71 | \( 1 + 7.17T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 - 4.66iT - 89T^{2} \) |
| 97 | \( 1 + 3.82iT - 97T^{2} \) |
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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
−9.444411615957849902174860255457, −8.591584522957761186257425833190, −7.75780598861123138180131619020, −6.94810310870929442501939023928, −5.86837524172682359741002793392, −4.74123109523784716445147550240, −4.18167207549052124229138580386, −2.47278051039369015064336065290, −1.57496718668128786031841974846, −0.47575819445438652828113327248,
2.35612711403107512679153491749, 3.53532617017366248311543314640, 4.37286802456165746174645776610, 5.19048633866360451089200351506, 6.44735323650968359595975643517, 6.98368200571582544607402845978, 7.69315438041585846648997431152, 9.024389386304608141017335951427, 9.680590891470037549172536832995, 10.12956947274664834432783855803