| L(s) = 1 | − 5.83i·5-s + 2.82·7-s + 16.4i·11-s − 8.24·13-s + 7.07i·17-s + 23.3·19-s + 20i·23-s − 9·25-s − 29.1i·29-s − 42.4·31-s − 16.4i·35-s − 49.4·37-s + 26.8i·41-s − 44i·47-s − 41·49-s + ⋯ |
| L(s) = 1 | − 1.16i·5-s + 0.404·7-s + 1.49i·11-s − 0.634·13-s + 0.415i·17-s + 1.22·19-s + 0.869i·23-s − 0.359·25-s − 1.00i·29-s − 1.36·31-s − 0.471i·35-s − 1.33·37-s + 0.655i·41-s − 0.936i·47-s − 0.836·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6538954744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6538954744\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.83iT - 25T^{2} \) |
| 7 | \( 1 - 2.82T + 49T^{2} \) |
| 11 | \( 1 - 16.4iT - 121T^{2} \) |
| 13 | \( 1 + 8.24T + 169T^{2} \) |
| 17 | \( 1 - 7.07iT - 289T^{2} \) |
| 19 | \( 1 - 23.3T + 361T^{2} \) |
| 23 | \( 1 - 20iT - 529T^{2} \) |
| 29 | \( 1 + 29.1iT - 841T^{2} \) |
| 31 | \( 1 + 42.4T + 961T^{2} \) |
| 37 | \( 1 + 49.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 44iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 29.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 65.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 82.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 116.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 100iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 40T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 82.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 114. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 40T + 9.40e3T^{2} \) |
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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
−9.208596212083745113816917310650, −8.305053669681277959620087151110, −7.55055762367132316024614637710, −7.02142414499553870755100854187, −5.70819891577149785513615964930, −5.03216338372843319511943754535, −4.51231592533944848677652684032, −3.46592905669838809814542398068, −2.04173468751360660635249274467, −1.33887922018307676792503958633,
0.15380743251030571359203949098, 1.59920914113754041541034629564, 3.03973472563440428719643733682, 3.18915253017141551879636002783, 4.58071875316883650521740405764, 5.51384912429169076716115270426, 6.18608061177118768872729578653, 7.24358201677243970370324596359, 7.48746877609516943464170111539, 8.679727339632582008374574036341
