| L(s) = 1 | − 1.73i·3-s + 9.91·5-s − 3.01·7-s − 2.99·9-s + 10.8·11-s + 15.4i·13-s − 17.1i·15-s + 0.115·17-s + (−1.92 + 18.9i)19-s + 5.22i·21-s + 26.9·23-s + 73.2·25-s + 5.19i·27-s − 34.0i·29-s + 27.7i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.98·5-s − 0.430·7-s − 0.333·9-s + 0.983·11-s + 1.18i·13-s − 1.14i·15-s + 0.00679·17-s + (−0.101 + 0.994i)19-s + 0.248i·21-s + 1.17·23-s + 2.93·25-s + 0.192i·27-s − 1.17i·29-s + 0.895i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.829853156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.829853156\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 19 | \( 1 + (1.92 - 18.9i)T \) |
| good | 5 | \( 1 - 9.91T + 25T^{2} \) |
| 7 | \( 1 + 3.01T + 49T^{2} \) |
| 11 | \( 1 - 10.8T + 121T^{2} \) |
| 13 | \( 1 - 15.4iT - 169T^{2} \) |
| 17 | \( 1 - 0.115T + 289T^{2} \) |
| 23 | \( 1 - 26.9T + 529T^{2} \) |
| 29 | \( 1 + 34.0iT - 841T^{2} \) |
| 31 | \( 1 - 27.7iT - 961T^{2} \) |
| 37 | \( 1 - 51.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 15.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 25.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 84.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 60.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.59iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 73.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 9.08T + 5.32e3T^{2} \) |
| 79 | \( 1 + 96.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 61.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 39.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 35.6iT - 9.40e3T^{2} \) |
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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
−9.733112161172478563294967961913, −9.207380135339253333365562563521, −8.424582071321209179181454197900, −6.84546834836226423294821562371, −6.55873719391252666089424607265, −5.77761011843512972633322627901, −4.73852498145052249829168325951, −3.24496163375330592877689334246, −2.01647817026711854881869276736, −1.32754332435617020365733878449,
1.06233929329203687854177642463, 2.44476868727391704136176524439, 3.33953559992929341311327153930, 4.83207145956977805829940165118, 5.54963339043087443329058820974, 6.32106734948095519705988716731, 7.10069751504034713610459521366, 8.664213117955765775088008832742, 9.272156587576039104867839115013, 9.770530376127636517503863089755
