| L(s) = 1 | − 3.46i·5-s + 2.64i·7-s − 2.92·11-s + 19.1i·13-s + 14.3·17-s − 8.09·19-s − 16.7i·23-s + 12.9·25-s − 27.1i·29-s + 44.8i·31-s + 9.16·35-s − 39.5i·37-s − 45.8·41-s − 61.0·43-s + 46.2i·47-s + ⋯ |
| L(s) = 1 | − 0.693i·5-s + 0.377i·7-s − 0.266·11-s + 1.47i·13-s + 0.846·17-s − 0.426·19-s − 0.728i·23-s + 0.519·25-s − 0.936i·29-s + 1.44i·31-s + 0.261·35-s − 1.06i·37-s − 1.11·41-s − 1.41·43-s + 0.984i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.036885087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036885087\) |
| \(L(2)\) |
|
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 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 + 3.46iT - 25T^{2} \) |
| 11 | \( 1 + 2.92T + 121T^{2} \) |
| 13 | \( 1 - 19.1iT - 169T^{2} \) |
| 17 | \( 1 - 14.3T + 289T^{2} \) |
| 19 | \( 1 + 8.09T + 361T^{2} \) |
| 23 | \( 1 + 16.7iT - 529T^{2} \) |
| 29 | \( 1 + 27.1iT - 841T^{2} \) |
| 31 | \( 1 - 44.8iT - 961T^{2} \) |
| 37 | \( 1 + 39.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 45.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 61.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 46.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 9.69iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 114.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 7.48iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 12.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 129. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 18.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 42.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 109.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 80.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 162.T + 9.40e3T^{2} \) |
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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
−8.996668177727142850834343925147, −8.647330941393157209203308932152, −7.73005833471794767462265730780, −6.79131303807154853172122967765, −6.08283767414149117435695427833, −5.04305105898795289777291970565, −4.50312892006487446557129571589, −3.40808190221764914847625008218, −2.23822600435032895747215210825, −1.23320069341065856239590454522,
0.26795131996062980569972746771, 1.62791304818724949077049542910, 3.04719623536023208327151413431, 3.40909335537205051798306060106, 4.75998223637605941951347197384, 5.53457334181535475806800510880, 6.38716564506038231731781343433, 7.24221991720706769839791080501, 7.88850047579267382466984507636, 8.560874537513054047875581589848
