L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − i·5-s + 0.999i·8-s + (0.5 + 0.866i)10-s + (1.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s − i·17-s + (−0.866 − 0.499i)20-s − 25-s − 1.73·26-s + (0.866 + 1.5i)29-s + (0.866 + 0.499i)32-s + (0.5 + 0.866i)34-s − 1.73i·37-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − i·5-s + 0.999i·8-s + (0.5 + 0.866i)10-s + (1.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s − i·17-s + (−0.866 − 0.499i)20-s − 25-s − 1.73·26-s + (0.866 + 1.5i)29-s + (0.866 + 0.499i)32-s + (0.5 + 0.866i)34-s − 1.73i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7915124002\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7915124002\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 1.73iT - T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + 2iT - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.73iT - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.217062409299484381038148794889, −8.819931056289019414985804490547, −8.183613562890395493117477577431, −7.18126467776400678656712914709, −6.47476685328454848412994025492, −5.53330195015241625009453675158, −4.82583962992809025588495286626, −3.65682689132238934106769251496, −2.06115072009436680961967416694, −0.984487932477870132049275227010,
1.32001599010555348249834510369, 2.63611711717537205240388623621, 3.39502238818422010992063958967, 4.27225547582750772383416098145, 6.08739896143301286395519270466, 6.30688976375522029569711531400, 7.52936608127299332023781692641, 8.097926223083776157902990126554, 8.792835803384186575845598242302, 9.798861136096596526831292679991