L(s) = 1 | + (0.0747 + 0.997i)4-s + (−0.222 + 0.974i)7-s + (−0.162 + 0.712i)13-s + (−0.988 + 0.149i)16-s + (0.222 − 0.385i)19-s + (−0.733 + 0.680i)25-s + (−0.988 − 0.149i)28-s + (−0.0747 − 0.129i)31-s + (0.0111 − 0.149i)37-s + (1.19 + 1.49i)43-s + (−0.900 − 0.433i)49-s + (−0.722 − 0.108i)52-s + (0.123 − 1.64i)61-s + (−0.222 − 0.974i)64-s + (0.988 + 1.71i)67-s + ⋯ |
L(s) = 1 | + (0.0747 + 0.997i)4-s + (−0.222 + 0.974i)7-s + (−0.162 + 0.712i)13-s + (−0.988 + 0.149i)16-s + (0.222 − 0.385i)19-s + (−0.733 + 0.680i)25-s + (−0.988 − 0.149i)28-s + (−0.0747 − 0.129i)31-s + (0.0111 − 0.149i)37-s + (1.19 + 1.49i)43-s + (−0.900 − 0.433i)49-s + (−0.722 − 0.108i)52-s + (0.123 − 1.64i)61-s + (−0.222 − 0.974i)64-s + (0.988 + 1.71i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9650966089\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9650966089\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.222 - 0.974i)T \) |
good | 2 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 5 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 11 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 13 | \( 1 + (0.162 - 0.712i)T + (-0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 19 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.0111 + 0.149i)T + (-0.988 - 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (-1.19 - 1.49i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 53 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 61 | \( 1 + (-0.123 + 1.64i)T + (-0.988 - 0.149i)T^{2} \) |
| 67 | \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-1.32 + 1.22i)T + (0.0747 - 0.997i)T^{2} \) |
| 79 | \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 97 | \( 1 - 1.91T + T^{2} \) |
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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
−9.734657709174157034235544277300, −9.232897913029808051548006225443, −8.449978820643668442303629419372, −7.67390138975381429611845220222, −6.84901626447994872351066157028, −5.98983480984804038034756631224, −4.94336860627420572620071287863, −3.94567773176881608575635814031, −2.96435512325056360074412012493, −2.05198414937931244931017363533,
0.821380339996888952947828142632, 2.19968774885279357180678175639, 3.56061507904456465480541524700, 4.52927006730274254081411325441, 5.49213496371417081139198326979, 6.23609286498714035631676994034, 7.13186061104414742010830892216, 7.86027748417921097735780622186, 8.929452935506238743584559261037, 9.843374813380338317128946635636