L(s) = 1 | − 2-s + 4-s + 2.23·5-s + (−2.23 + 1.41i)7-s − 8-s − 2.23·10-s − 5.65i·11-s + 4.47·13-s + (2.23 − 1.41i)14-s + 16-s − 3.16i·17-s + 3.16i·19-s + 2.23·20-s + 5.65i·22-s + 4·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.999·5-s + (−0.845 + 0.534i)7-s − 0.353·8-s − 0.707·10-s − 1.70i·11-s + 1.24·13-s + (0.597 − 0.377i)14-s + 0.250·16-s − 0.766i·17-s + 0.725i·19-s + 0.499·20-s + 1.20i·22-s + 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20353 - 0.238568i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20353 - 0.238568i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 + (2.23 - 1.41i)T \) |
good | 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 3.16iT - 17T^{2} \) |
| 19 | \( 1 - 3.16iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 9.89iT - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 1.41iT - 43T^{2} \) |
| 47 | \( 1 - 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 9.48iT - 61T^{2} \) |
| 67 | \( 1 + 7.07iT - 67T^{2} \) |
| 71 | \( 1 + 1.41iT - 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 12.6iT - 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.60158489171529495054947567698, −9.405702175972183013338602970666, −9.012617331621855493313417264647, −8.210782543070990292402841478554, −6.84645753778366099551293349394, −6.01478968443605993822346958506, −5.54423269449902145548742758426, −3.52979598309211394140758551426, −2.62350898074902184094810106506, −1.00874170042275926146526637683,
1.31849549686020512285836631897, 2.57075342468429059742407286296, 3.96575380759150427274872682529, 5.30855532129483966554312606387, 6.56970317365864218330335060145, 6.84929622828197415809552061831, 8.126659317883966530613513668424, 9.181782775284915260591642030126, 9.734590223846906067398522202255, 10.42100354076787456095611536295