L(s) = 1 | − i·7-s + 4i·13-s − 6i·17-s − 2·19-s + 3i·23-s − 3·29-s − 10·31-s − 10i·37-s + 9·41-s + 4i·43-s + 9i·47-s + 6·49-s + 6i·53-s + 6·59-s − 61-s + ⋯ |
L(s) = 1 | − 0.377i·7-s + 1.10i·13-s − 1.45i·17-s − 0.458·19-s + 0.625i·23-s − 0.557·29-s − 1.79·31-s − 1.64i·37-s + 1.40·41-s + 0.609i·43-s + 1.31i·47-s + 0.857·49-s + 0.824i·53-s + 0.781·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.434637800\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.434637800\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 11iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−7.65408368602153946910317893783, −7.34856309410404754744237377889, −6.71222020256871967185726484802, −5.79008505536397935651901477441, −5.23248344226306672608431661306, −4.22946755459864892833013698681, −3.87027439297491784311532031452, −2.72863547557977660385535780707, −1.98642216765270683645099395007, −0.878237822811585259886704335801,
0.39795018427201534517092136354, 1.69599074879575490912155350018, 2.44455112175389258535846772273, 3.47497600246573934504818003249, 3.99121276787776724876234885927, 5.05655537843348592630617836949, 5.61626185623178415255458117535, 6.26442757525536965973921745495, 6.97931461442362710704117522986, 7.85968408716017046476790503943