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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.199539009\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.199539009\) |
\(L(1)\) |
|
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
−11.07592057038390060162940716900, −10.05586980511069508266395755154, −9.241547697533667662313922992920, −8.362875669837112914179179469359, −7.80320660940018841398100014026, −6.61865988591937413321884178296, −5.51026403136219410178895749480, −3.83943903079494215736002750420, −3.54804008589239860115726137883, −2.37096183055392800921739797465,
1.63201082165071995425070317774, 2.75816156429028855192988974024, 4.12899350616390728545160209936, 4.96821055562375479080541028448, 6.57776311551622946290530084431, 7.44449531525485799429013735646, 8.171664173808442870983772355463, 8.813405943952122203547427710859, 9.644814345179979862396551452419, 10.64954108022958081589778521906