| L(s) = 1 | + 9.10i·5-s + (−0.958 − 6.93i)7-s − 9.66i·11-s + 2.31·13-s + 21.8·17-s + 26.9i·19-s − 15.4·23-s − 57.9·25-s − 18.1·29-s − 45.9·31-s + (63.1 − 8.73i)35-s + 32.3i·37-s + 33.0·41-s + 60.5·43-s + 29.4i·47-s + ⋯ |
| L(s) = 1 | + 1.82i·5-s + (−0.136 − 0.990i)7-s − 0.878i·11-s + 0.178·13-s + 1.28·17-s + 1.41i·19-s − 0.670·23-s − 2.31·25-s − 0.625·29-s − 1.48·31-s + (1.80 − 0.249i)35-s + 0.873i·37-s + 0.806·41-s + 1.40·43-s + 0.627i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9636305789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9636305789\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.958 + 6.93i)T \) |
| good | 5 | \( 1 - 9.10iT - 25T^{2} \) |
| 11 | \( 1 + 9.66iT - 121T^{2} \) |
| 13 | \( 1 - 2.31T + 169T^{2} \) |
| 17 | \( 1 - 21.8T + 289T^{2} \) |
| 19 | \( 1 - 26.9iT - 361T^{2} \) |
| 23 | \( 1 + 15.4T + 529T^{2} \) |
| 29 | \( 1 + 18.1T + 841T^{2} \) |
| 31 | \( 1 + 45.9T + 961T^{2} \) |
| 37 | \( 1 - 32.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 33.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 60.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 29.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 65.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 8.47T + 3.48e3T^{2} \) |
| 61 | \( 1 + 74.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 20.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 21.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 100. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 78.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 67.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 130.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 175. iT - 9.40e3T^{2} \) |
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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
−9.640523931974869499421003979560, −8.288767954650446098557132982454, −7.52948366184725595860443210487, −7.16215596189296138121986838525, −6.00905049843365030533501217947, −5.78006517260331929819445346725, −3.96009437243115934827242313701, −3.57801548828665470887981982148, −2.71070397442132521240619977468, −1.35087080699882175350582603326,
0.24558293881127937971220686806, 1.46931717075958590154012959056, 2.40444096586384172282235630501, 3.82005685911471872880800478591, 4.66443003940326679032916835365, 5.45750561590563397301845488691, 5.85733802767434085812317376414, 7.28045141745289374065978346666, 7.901132674606086252703878547808, 8.948932050232932812434660870461
