L(s) = 1 | + i·2-s − 1.36i·3-s − 4-s + (1.38 + 1.75i)5-s + 1.36·6-s + 0.636i·7-s − i·8-s + 1.14·9-s + (−1.75 + 1.38i)10-s + 3.50·11-s + 1.36i·12-s + 0.141i·13-s − 0.636·14-s + (2.38 − 1.89i)15-s + 16-s + 2.14i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.787i·3-s − 0.5·4-s + (0.621 + 0.783i)5-s + 0.556·6-s + 0.240i·7-s − 0.353i·8-s + 0.380·9-s + (−0.554 + 0.439i)10-s + 1.05·11-s + 0.393i·12-s + 0.0391i·13-s − 0.170·14-s + (0.616 − 0.488i)15-s + 0.250·16-s + 0.519i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21683 + 0.423776i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21683 + 0.423776i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-1.38 - 1.75i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 1.36iT - 3T^{2} \) |
| 7 | \( 1 - 0.636iT - 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 - 0.141iT - 13T^{2} \) |
| 17 | \( 1 - 2.14iT - 17T^{2} \) |
| 23 | \( 1 + 4.91iT - 23T^{2} \) |
| 29 | \( 1 + 7.15T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 - 3.27iT - 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 + 2.49iT - 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 + 8.14iT - 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 - 8.37iT - 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 - 3.69iT - 73T^{2} \) |
| 79 | \( 1 - 4.17T + 79T^{2} \) |
| 83 | \( 1 - 9.00iT - 83T^{2} \) |
| 89 | \( 1 - 6.77T + 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−12.90356667263701116500892765978, −11.87078147064312528679265012519, −10.60255783512724041641928729540, −9.555164455432317290604680722520, −8.503655285181399809546950927886, −7.17215960297776972529262274790, −6.63990066743912451584119523402, −5.62401297475751619080272198727, −3.86729568264320067947768012528, −1.90255614503579768851563293398,
1.59486006614633122519785379543, 3.66373112377009733709320178132, 4.62643853548956378579857939981, 5.77390626129074940428910387873, 7.43090163882148234392101774312, 9.198291178610722541948875229885, 9.289398527533094864208131855368, 10.42791842075972560550982798719, 11.38056419530427717937753207683, 12.45355914397413285174338145396