| L(s) = 1 | − 2.54i·3-s + 4.11i·5-s + 2.54·9-s − 3.26·11-s + 5.88i·13-s + 10.4·15-s + 13.8i·17-s + 15.8i·19-s − 36.5·23-s + 8.03·25-s − 29.3i·27-s − 28.4·29-s − 41.8i·31-s + 8.30i·33-s + 14.2·37-s + ⋯ |
| L(s) = 1 | − 0.847i·3-s + 0.823i·5-s + 0.282·9-s − 0.297·11-s + 0.452i·13-s + 0.697·15-s + 0.816i·17-s + 0.832i·19-s − 1.58·23-s + 0.321·25-s − 1.08i·27-s − 0.981·29-s − 1.34i·31-s + 0.251i·33-s + 0.386·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5265054976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5265054976\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.54iT - 9T^{2} \) |
| 5 | \( 1 - 4.11iT - 25T^{2} \) |
| 11 | \( 1 + 3.26T + 121T^{2} \) |
| 13 | \( 1 - 5.88iT - 169T^{2} \) |
| 17 | \( 1 - 13.8iT - 289T^{2} \) |
| 19 | \( 1 - 15.8iT - 361T^{2} \) |
| 23 | \( 1 + 36.5T + 529T^{2} \) |
| 29 | \( 1 + 28.4T + 841T^{2} \) |
| 31 | \( 1 + 41.8iT - 961T^{2} \) |
| 37 | \( 1 - 14.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 21.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 33.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 84.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 67.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 29.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 54.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 83.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 125. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 70.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 27.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 11.3iT - 9.40e3T^{2} \) |
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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
−9.833594280996914228160140192759, −8.498202189320124554680652169628, −7.81823331760674495684710619773, −7.20306128417437475289642152892, −6.32605247462824445526544788580, −5.84334092112054518517020363574, −4.39755807441123110451522979795, −3.56303724543042160383719214412, −2.30247284373290439008544075098, −1.55281406061478563229090674004,
0.13680693687915118089167008499, 1.55237215918478175295515354163, 2.95903537502894830318669738751, 3.95915040482383652886753899507, 4.87741594378096347692883668548, 5.25324536874942873179099066932, 6.46162169735563850784030245104, 7.45221223879325751945600959468, 8.254728269649173157309461268200, 9.126867885161734235231775956231
