L(s) = 1 | + 1.41i·5-s + i·7-s − 5.65·11-s + 4·13-s + 1.41i·17-s + 8i·19-s + 5.65·23-s + 2.99·25-s − 7.07i·29-s − 1.41·35-s − 2·37-s − 7.07i·41-s + 8i·43-s − 49-s + 9.89i·53-s + ⋯ |
L(s) = 1 | + 0.632i·5-s + 0.377i·7-s − 1.70·11-s + 1.10·13-s + 0.342i·17-s + 1.83i·19-s + 1.17·23-s + 0.599·25-s − 1.31i·29-s − 0.239·35-s − 0.328·37-s − 1.10i·41-s + 1.21i·43-s − 0.142·49-s + 1.35i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.105308596\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.105308596\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 7.07iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 9.89iT - 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−8.593611128363273601224452939354, −7.958831279293771034407919179539, −7.45303302139967765016531532892, −6.36272238025537020729251035944, −5.86816055360073105199745162782, −5.13707244051449764264724347845, −4.09799517261358728488578740385, −3.18147701850301200358326062135, −2.53869012599120926090535140241, −1.37315740377924290785141313479,
0.32765289180548525008186328952, 1.37501909149461390881464938782, 2.73563106964815867096470403683, 3.31539933004828893363612174500, 4.71263140818280089422446925078, 4.93411843854818119853245464653, 5.78677873752171513498843042968, 6.91465595655528596375865943110, 7.27941552508155516733687146342, 8.365217506766491579371040117036