| L(s) = 1 | + 2.23i·5-s + (1.58 − 2.12i)7-s − 1.41i·11-s − 3.16·13-s + 4.47i·17-s − 6·23-s − 5.00·25-s − 2.82i·29-s + (4.74 + 3.53i)35-s − 4.24i·37-s + 9.48·41-s − 8.48i·43-s − 4.47i·47-s + (−1.99 − 6.70i)49-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.999i·5-s + (0.597 − 0.801i)7-s − 0.426i·11-s − 0.877·13-s + 1.08i·17-s − 1.25·23-s − 1.00·25-s − 0.525i·29-s + (0.801 + 0.597i)35-s − 0.697i·37-s + 1.48·41-s − 1.29i·43-s − 0.652i·47-s + (−0.285 − 0.958i)49-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.110962806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.110962806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.23iT \) |
| 7 | \( 1 + (-1.58 + 2.12i)T \) |
| good | 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 4.24iT - 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 4.47iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 6.32T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.81629357957584422348474008407, −7.50430279277827520454523090335, −6.61909863224997141236779611414, −6.00043497815475217054996560258, −5.16919253651384196772693673801, −4.06549618961220356124573713906, −3.73101046102581512856933256568, −2.52422043252391333721802079431, −1.80153233270709039308276910238, −0.30261590911532246391102794893,
1.13926291074774644563276698896, 2.14047776080201538539135117748, 2.88396492585357174144512459798, 4.29764279235339515043763887730, 4.69230235386863609964839588787, 5.45354199936792343790444030549, 6.04588769520967230110993180667, 7.15480875234800300211507369614, 7.81377405762770609758255196927, 8.351674825447241681574261738736
