| L(s) = 1 | + 1.73·3-s − 4.05i·5-s − 2.64i·7-s + 2.99·9-s + 11.2i·11-s + 2.33·13-s − 7.02i·15-s + 29.3i·17-s − 23.1i·19-s − 4.58i·21-s + (11.4 − 19.9i)23-s + 8.54·25-s + 5.19·27-s + 12.9·29-s − 10.0·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.811i·5-s − 0.377i·7-s + 0.333·9-s + 1.02i·11-s + 0.179·13-s − 0.468i·15-s + 1.72i·17-s − 1.22i·19-s − 0.218i·21-s + (0.498 − 0.867i)23-s + 0.341·25-s + 0.192·27-s + 0.447·29-s − 0.324·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.682018694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.682018694\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 7 | \( 1 + 2.64iT \) |
| 23 | \( 1 + (-11.4 + 19.9i)T \) |
| good | 5 | \( 1 + 4.05iT - 25T^{2} \) |
| 11 | \( 1 - 11.2iT - 121T^{2} \) |
| 13 | \( 1 - 2.33T + 169T^{2} \) |
| 17 | \( 1 - 29.3iT - 289T^{2} \) |
| 19 | \( 1 + 23.1iT - 361T^{2} \) |
| 29 | \( 1 - 12.9T + 841T^{2} \) |
| 31 | \( 1 + 10.0T + 961T^{2} \) |
| 37 | \( 1 - 9.47iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 62.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 8.51iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 68.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 50.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 41.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 27.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 23.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 94.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 24.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 8.80iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 29.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 159. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 153. iT - 9.40e3T^{2} \) |
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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
−8.899624925499713218985455636799, −8.319058583702180388387466740046, −7.43082712242545352279783424799, −6.73192841344478810685680190675, −5.72173254457089148567296129908, −4.51408665118432065820002678574, −4.28681499721664623090738961626, −2.95481991594061270940551310446, −1.88829058225312558190433364681, −0.830447803203802633520662631424,
0.938148978094122439071225838424, 2.40065224356012821423815636633, 3.09466579437327239275622260715, 3.84920898911090618348247332552, 5.13586447240636117884406150110, 5.91055794079933739522631332643, 6.81983328887874127175818701989, 7.54685257430306060256954343815, 8.265112808486916561773701083768, 9.170032012814243085513155321845
