L(s) = 1 | + (−0.363 − 0.5i)3-s + (0.190 − 0.587i)9-s + (0.587 + 0.809i)11-s + (−0.5 − 1.53i)17-s + (0.951 + 1.30i)19-s + (0.809 − 0.587i)25-s + (−0.951 + 0.309i)27-s + (0.190 − 0.587i)33-s + (0.5 − 0.363i)41-s − 0.618i·43-s + (0.309 + 0.951i)49-s + (−0.587 + 0.809i)51-s + (0.309 − 0.951i)57-s + (0.951 − 1.30i)59-s − 1.61i·67-s + ⋯ |
L(s) = 1 | + (−0.363 − 0.5i)3-s + (0.190 − 0.587i)9-s + (0.587 + 0.809i)11-s + (−0.5 − 1.53i)17-s + (0.951 + 1.30i)19-s + (0.809 − 0.587i)25-s + (−0.951 + 0.309i)27-s + (0.190 − 0.587i)33-s + (0.5 − 0.363i)41-s − 0.618i·43-s + (0.309 + 0.951i)49-s + (−0.587 + 0.809i)51-s + (0.309 − 0.951i)57-s + (0.951 − 1.30i)59-s − 1.61i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.142701155\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.142701155\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (-0.587 - 0.809i)T \) |
good | 3 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + 0.618iT - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + 1.61iT - T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 0.618T + T^{2} \) |
| 97 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.036463288916524052422143308024, −7.953791168691187555661484927658, −7.18158786852728436217618135245, −6.74919734986664379145336472511, −5.89587142708707045667907735243, −5.03146318500948112379403063600, −4.15897390591405130318172172743, −3.21828601996219833552112537357, −2.03587572818094307007747942992, −0.916236904739694247434245719289,
1.24054915449849169087308295044, 2.55310026542305238196571374968, 3.62094783270972796528766668595, 4.39636456110711061458654572053, 5.22027280169041745715354945723, 5.92856256932458630384681509770, 6.77744573682330167348706633703, 7.54344313949232591825203523556, 8.523665750692238107623802558185, 8.984389005992329607270361613078