L(s) = 1 | + (−0.826 + 0.563i)4-s + (0.733 + 0.680i)7-s + (−0.548 − 1.77i)13-s + (0.365 − 0.930i)16-s + (−1.68 − 0.974i)19-s + (−0.222 − 0.974i)25-s + (−0.988 − 0.149i)28-s + (−1.72 − 0.997i)31-s + (−0.0111 + 0.149i)37-s + (1.88 − 0.284i)43-s + (0.0747 + 0.997i)49-s + (1.45 + 1.16i)52-s + (−0.634 + 0.930i)61-s + (0.222 + 0.974i)64-s + (0.988 − 1.71i)67-s + ⋯ |
L(s) = 1 | + (−0.826 + 0.563i)4-s + (0.733 + 0.680i)7-s + (−0.548 − 1.77i)13-s + (0.365 − 0.930i)16-s + (−1.68 − 0.974i)19-s + (−0.222 − 0.974i)25-s + (−0.988 − 0.149i)28-s + (−1.72 − 0.997i)31-s + (−0.0111 + 0.149i)37-s + (1.88 − 0.284i)43-s + (0.0747 + 0.997i)49-s + (1.45 + 1.16i)52-s + (−0.634 + 0.930i)61-s + (0.222 + 0.974i)64-s + (0.988 − 1.71i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7070736129\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7070736129\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.733 - 0.680i)T \) |
good | 2 | \( 1 + (0.826 - 0.563i)T^{2} \) |
| 5 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 13 | \( 1 + (0.548 + 1.77i)T + (-0.826 + 0.563i)T^{2} \) |
| 17 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 19 | \( 1 + (1.68 + 0.974i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 29 | \( 1 + (-0.988 - 0.149i)T^{2} \) |
| 31 | \( 1 + (1.72 + 0.997i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.0111 - 0.149i)T + (-0.988 - 0.149i)T^{2} \) |
| 41 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 43 | \( 1 + (-1.88 + 0.284i)T + (0.955 - 0.294i)T^{2} \) |
| 47 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 53 | \( 1 + (-0.988 + 0.149i)T^{2} \) |
| 59 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 61 | \( 1 + (0.634 - 0.930i)T + (-0.365 - 0.930i)T^{2} \) |
| 67 | \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.590 - 0.636i)T + (-0.0747 + 0.997i)T^{2} \) |
| 79 | \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 89 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 97 | \( 1 + (0.510 + 0.294i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.447810363261819957185794373965, −7.87593751711261304374451374631, −7.28876812114102865956090310954, −6.03985938553685249141551027999, −5.42919204625817911875624095498, −4.66708286220936518198299218439, −4.01295893249536489892864337893, −2.88201703983541939648405821673, −2.20162142737553422307638610213, −0.40642214940556207912272534927,
1.43273452004547430724179965067, 2.08727851187436172123223841588, 3.86905793139905928610812227962, 4.15487250057929968041000197827, 4.97607449812258945588712281220, 5.72583956002042645031337516156, 6.65166648832846160661141679714, 7.30695505173448683512374666206, 8.168750383457153295676704713968, 8.898979104187966264994258691460