L(s) = 1 | + (−1.11 + 0.642i)7-s + (1.43 − 1.20i)13-s + (0.5 + 0.866i)19-s + (−0.766 + 0.642i)25-s + (1.70 − 0.984i)31-s + 0.347·37-s + (0.673 + 1.85i)43-s + (0.326 − 0.565i)49-s + (1.76 + 0.642i)61-s + (−1.26 + 0.223i)67-s + (1.43 + 1.20i)73-s + (0.826 − 0.984i)79-s + (−0.826 + 2.27i)91-s + (0.173 − 0.984i)97-s + (0.592 + 0.342i)103-s + ⋯ |
L(s) = 1 | + (−1.11 + 0.642i)7-s + (1.43 − 1.20i)13-s + (0.5 + 0.866i)19-s + (−0.766 + 0.642i)25-s + (1.70 − 0.984i)31-s + 0.347·37-s + (0.673 + 1.85i)43-s + (0.326 − 0.565i)49-s + (1.76 + 0.642i)61-s + (−1.26 + 0.223i)67-s + (1.43 + 1.20i)73-s + (0.826 − 0.984i)79-s + (−0.826 + 2.27i)91-s + (0.173 − 0.984i)97-s + (0.592 + 0.342i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.147190261\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147190261\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 0.347T + T^{2} \) |
| 41 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (1.26 - 0.223i)T + (0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.826 + 0.984i)T + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
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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
−9.116814315100033415387444217183, −8.185304952829958738767130687393, −7.76975555812698937213727644017, −6.50983212179550317256604588653, −5.99489402740511512980515197594, −5.46276381312502239644359543023, −4.12869817702308375810093684644, −3.33577941178577100644431955216, −2.62515174892172630128278269246, −1.11395685805256252589687116167,
0.946199370142556663411489282118, 2.34519510248242047685385955893, 3.48997039377118444582394382187, 4.02607581809325126458396775564, 5.01750793759783187173446403523, 6.22595404700504401700355054804, 6.55902798247310263754352816820, 7.30049972245613497630059192622, 8.333981722550018147799886105768, 8.995427324468948786173414461848