L(s) = 1 | − 3.08e3i·3-s − 3.49e4i·5-s + (7.75e5 − 2.76e5i)7-s − 4.76e6·9-s − 1.57e7·11-s − 7.52e7i·13-s − 1.07e8·15-s − 5.31e8i·17-s + 6.79e8i·19-s + (−8.53e8 − 2.39e9i)21-s + 2.16e9·23-s − 1.22e9·25-s − 5.62e7i·27-s − 3.70e9·29-s − 3.57e9i·31-s + ⋯ |
L(s) = 1 | − 1.41i·3-s − 0.447i·5-s + (0.942 − 0.335i)7-s − 0.996·9-s − 0.807·11-s − 1.19i·13-s − 0.631·15-s − 1.29i·17-s + 0.760i·19-s + (−0.473 − 1.33i)21-s + 0.634·23-s − 0.199·25-s − 0.00537i·27-s − 0.214·29-s − 0.129i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(1.336835244\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.336835244\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 3.49e4iT \) |
| 7 | \( 1 + (-7.75e5 + 2.76e5i)T \) |
good | 3 | \( 1 + 3.08e3iT - 4.78e6T^{2} \) |
| 11 | \( 1 + 1.57e7T + 3.79e14T^{2} \) |
| 13 | \( 1 + 7.52e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + 5.31e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 6.79e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 - 2.16e9T + 1.15e19T^{2} \) |
| 29 | \( 1 + 3.70e9T + 2.97e20T^{2} \) |
| 31 | \( 1 + 3.57e9iT - 7.56e20T^{2} \) |
| 37 | \( 1 - 5.03e9T + 9.01e21T^{2} \) |
| 41 | \( 1 - 2.33e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 2.46e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 4.68e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 1.75e11T + 1.37e24T^{2} \) |
| 59 | \( 1 + 3.59e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 3.47e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 + 5.66e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 7.81e10T + 8.27e25T^{2} \) |
| 73 | \( 1 - 7.35e12iT - 1.22e26T^{2} \) |
| 79 | \( 1 + 2.41e13T + 3.68e26T^{2} \) |
| 83 | \( 1 - 7.63e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 + 1.57e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 - 6.35e13iT - 6.52e27T^{2} \) |
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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
−9.896941468288032139158644098989, −8.336892780707373647791452344703, −7.83723258824294672566268777074, −6.99781136989533162855810877540, −5.63071912147599155040008116224, −4.80778035687504440612113374423, −3.07130814317196598289570948481, −1.94778993896917799321834286664, −1.01487181407838478739747565419, −0.26092083434820886548470582979,
1.63071330459399913818046695994, 2.80058596088372328875153144879, 4.04530446956561968880624171898, 4.77668486725455012121501375516, 5.77840295818668902137508128764, 7.20376496826871471407647072646, 8.501613142262367805869334123505, 9.263632546595104745786467932706, 10.46018914030753059899597115971, 10.93568523172066346021630452313