L(s) = 1 | − 1.64·3-s − 4.02i·5-s − i·7-s − 0.305·9-s + i·11-s + (3.58 + 0.341i)13-s + 6.61i·15-s + 2.93·17-s − 5.26i·19-s + 1.64i·21-s − 6.16·23-s − 11.2·25-s + 5.42·27-s + 2.51·29-s − 9.67i·31-s + ⋯ |
L(s) = 1 | − 0.947·3-s − 1.80i·5-s − 0.377i·7-s − 0.101·9-s + 0.301i·11-s + (0.995 + 0.0946i)13-s + 1.70i·15-s + 0.712·17-s − 1.20i·19-s + 0.358i·21-s − 1.28·23-s − 2.24·25-s + 1.04·27-s + 0.466·29-s − 1.73i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9007564324\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9007564324\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-3.58 - 0.341i)T \) |
good | 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 + 4.02iT - 5T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 + 5.26iT - 19T^{2} \) |
| 23 | \( 1 + 6.16T + 23T^{2} \) |
| 29 | \( 1 - 2.51T + 29T^{2} \) |
| 31 | \( 1 + 9.67iT - 31T^{2} \) |
| 37 | \( 1 + 0.752iT - 37T^{2} \) |
| 41 | \( 1 + 2.05iT - 41T^{2} \) |
| 43 | \( 1 - 3.54T + 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 - 7.99T + 53T^{2} \) |
| 59 | \( 1 + 3.00iT - 59T^{2} \) |
| 61 | \( 1 - 6.61T + 61T^{2} \) |
| 67 | \( 1 + 8.37iT - 67T^{2} \) |
| 71 | \( 1 - 7.73iT - 71T^{2} \) |
| 73 | \( 1 + 6.96iT - 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 13.4iT - 83T^{2} \) |
| 89 | \( 1 - 12.5iT - 89T^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−8.239976258713757936563210994257, −7.38865176838995852800226020691, −6.35086498873835360046924605170, −5.71295393088186642066257999114, −5.16714432439848910795327597226, −4.38883600142865350344421392034, −3.77550870558990265663274465944, −2.19575882581420532396710855102, −1.01849011405082245746873676770, −0.36420192333410913741039371169,
1.38076473464356170589043462608, 2.68449316703855279296133779130, 3.32732225273110444964077994261, 4.14606481170001080999200382577, 5.55114994670910690703219803136, 5.89947704749484585291561542712, 6.44811639785329015407816300637, 7.16878310272426082929315855246, 8.062548513039465716118062044434, 8.635607916199699449946503567084