| L(s) = 1 | + 27i·3-s + (−98.1 − 261. i)5-s + 940. i·7-s − 729·9-s + 8.39e3·11-s + 1.46e3i·13-s + (7.06e3 − 2.64e3i)15-s − 1.99e4i·17-s − 3.68e4·19-s − 2.54e4·21-s − 1.40e4i·23-s + (−5.88e4 + 5.13e4i)25-s − 1.96e4i·27-s + 4.03e4·29-s − 1.93e3·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.351 − 0.936i)5-s + 1.03i·7-s − 0.333·9-s + 1.90·11-s + 0.184i·13-s + (0.540 − 0.202i)15-s − 0.982i·17-s − 1.23·19-s − 0.598·21-s − 0.240i·23-s + (−0.753 + 0.657i)25-s − 0.192i·27-s + 0.306·29-s − 0.0116·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.064258226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064258226\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27iT \) |
| 5 | \( 1 + (98.1 + 261. i)T \) |
| good | 7 | \( 1 - 940. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 8.39e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.46e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.99e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 3.68e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.40e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 4.03e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.93e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.29e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 1.96e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.18e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 9.66e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.45e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.68e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 6.87e3T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.21e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.81e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.64e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 6.45e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.35e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 6.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.12e6iT - 8.07e13T^{2} \) |
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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
−11.13231078177887200636291039547, −9.628156378463281523815862651319, −9.007736775013983204757894550510, −8.409735799768775333495151411354, −6.81649242178145998090806146512, −5.72043632219702102062730519888, −4.61505407845621830264642102698, −3.77367723909385808210467752239, −2.17829212241446208370069088885, −0.75908558559869007653224324848,
0.74239502260218737686862045155, 1.92486815001515825854223174201, 3.51798459347805759509661574932, 4.22838376684140554942289984084, 6.25368586466741361368287363986, 6.71680214989747569076492150152, 7.69689186158239800398759402002, 8.734664793560785287567813908572, 10.04064537116393162852041195913, 10.90885294568061906536895598060
