L(s) = 1 | + (0.222 + 0.974i)3-s + (−0.900 − 0.433i)7-s + (−0.900 + 0.433i)9-s + (−0.846 + 1.75i)13-s − 1.80·19-s + (0.222 − 0.974i)21-s + (0.900 − 0.433i)25-s + (−0.623 − 0.781i)27-s − 1.24·31-s + (−0.777 + 0.974i)37-s + (−1.90 − 0.433i)39-s + (0.846 + 0.193i)43-s + (0.623 + 0.781i)49-s + (−0.400 − 1.75i)57-s + (−1.22 − 0.974i)61-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)3-s + (−0.900 − 0.433i)7-s + (−0.900 + 0.433i)9-s + (−0.846 + 1.75i)13-s − 1.80·19-s + (0.222 − 0.974i)21-s + (0.900 − 0.433i)25-s + (−0.623 − 0.781i)27-s − 1.24·31-s + (−0.777 + 0.974i)37-s + (−1.90 − 0.433i)39-s + (0.846 + 0.193i)43-s + (0.623 + 0.781i)49-s + (−0.400 − 1.75i)57-s + (−1.22 − 0.974i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5403813714\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5403813714\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.222 - 0.974i)T \) |
| 7 | \( 1 + (0.900 + 0.433i)T \) |
good | 5 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 13 | \( 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 29 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + 1.24T + T^{2} \) |
| 37 | \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 41 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (-0.846 - 0.193i)T + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 53 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 61 | \( 1 + (1.22 + 0.974i)T + (0.222 + 0.974i)T^{2} \) |
| 67 | \( 1 - 1.94iT - T^{2} \) |
| 71 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.678 - 1.40i)T + (-0.623 + 0.781i)T^{2} \) |
| 79 | \( 1 - 1.56iT - T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 97 | \( 1 + 0.867iT - T^{2} \) |
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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
−9.461020715979829993653182569608, −8.983635679787533028317400812394, −8.230144390955337009129655463962, −6.99414485577534931986118863646, −6.60262387547900070002598268357, −5.53796184643977966561596725028, −4.41137760130188995991913336089, −4.14566040655343131505209510875, −3.00572595977367940992522778050, −2.06139177548181627442371401479,
0.32137833805072084174409341947, 2.02800284648896336199436471169, 2.84332239119603107924381601805, 3.61492148778531330229414223324, 5.05640584343658238903604878671, 5.83891433617304981860410593220, 6.48275693138313535966865563762, 7.34618593301765261759733308366, 7.900716488768856780534498607909, 8.881429499211967658479214454989