L(s) = 1 | + (−1.22 + 0.707i)3-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)9-s − 1.41i·11-s + (1.22 + 0.707i)15-s + (0.5 − 0.866i)17-s + (1.22 − 0.707i)19-s + 1.41i·23-s − 29-s − 1.41i·31-s + (1.00 + 1.73i)33-s + (0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s − 0.999·45-s − 1.41i·47-s + ⋯ |
L(s) = 1 | + (−1.22 + 0.707i)3-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)9-s − 1.41i·11-s + (1.22 + 0.707i)15-s + (0.5 − 0.866i)17-s + (1.22 − 0.707i)19-s + 1.41i·23-s − 29-s − 1.41i·31-s + (1.00 + 1.73i)33-s + (0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s − 0.999·45-s − 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5241667934\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5241667934\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−11.05779467564340188764339979834, −9.886636880387940706120462679483, −9.220529614544314714384674865838, −8.176022349827296675616119761690, −7.18860912721315605176890902357, −5.72870039958825678645468658696, −5.40722337466121424923603247001, −4.37084575160342456129979106339, −3.27488199213601075280953423016, −0.76938995368857870500129902383,
1.64808955771919695423046612081, 3.30605257888081308897905630880, 4.65120440150535555803883280940, 5.70139342786165549215124023856, 6.63872810897982195915192396304, 7.24945526794015763433665512611, 7.997131001163354181015222975757, 9.508771912589612026799186133461, 10.46235993358756787294598719659, 11.01807674991680858609077643429