L(s) = 1 | + 3-s + 3.19i·5-s + i·7-s + 9-s + 5.06i·11-s + (−1.56 + 3.24i)13-s + 3.19i·15-s + 0.320·17-s + 0.621i·19-s + i·21-s + 4.19·23-s − 5.19·25-s + 27-s − 2.30·29-s − 0.740i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.42i·5-s + 0.377i·7-s + 0.333·9-s + 1.52i·11-s + (−0.434 + 0.900i)13-s + 0.824i·15-s + 0.0777·17-s + 0.142i·19-s + 0.218i·21-s + 0.874·23-s − 1.03·25-s + 0.192·27-s − 0.427·29-s − 0.133i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.035583570\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.035583570\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (1.56 - 3.24i)T \) |
good | 5 | \( 1 - 3.19iT - 5T^{2} \) |
| 11 | \( 1 - 5.06iT - 11T^{2} \) |
| 17 | \( 1 - 0.320T + 17T^{2} \) |
| 19 | \( 1 - 0.621iT - 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 + 2.30T + 29T^{2} \) |
| 31 | \( 1 + 0.740iT - 31T^{2} \) |
| 37 | \( 1 + 7.42iT - 37T^{2} \) |
| 41 | \( 1 - 9.49iT - 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 4.98iT - 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 + 0.680iT - 59T^{2} \) |
| 61 | \( 1 + 3.13T + 61T^{2} \) |
| 67 | \( 1 - 13.3iT - 67T^{2} \) |
| 71 | \( 1 + 13.0iT - 71T^{2} \) |
| 73 | \( 1 + 14.2iT - 73T^{2} \) |
| 79 | \( 1 + 3.55T + 79T^{2} \) |
| 83 | \( 1 - 9.30iT - 83T^{2} \) |
| 89 | \( 1 + 13.6iT - 89T^{2} \) |
| 97 | \( 1 + 4.73iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.785315872899265090187261467199, −7.56053800579715968470903447666, −7.39730586860629491556170258360, −6.67150042469522902142118193375, −5.93679252407093197573674282305, −4.76577236487544335572236857213, −4.13742471086375017497617981123, −3.09951937763396922693645330057, −2.44864670179853595810420766013, −1.72407757454616935995952449787,
0.54066370389451240197192166366, 1.24825021437426138751784126651, 2.61333564658528531388995483925, 3.45128974602815725080974474581, 4.23449997689169156861555978562, 5.19012719217652983263777968236, 5.57747952812829986579427266097, 6.64008259819508023292440896243, 7.61474478943971102227987926050, 8.135109697842332066932813123464