| L(s) = 1 | + (−4.73e5 + 2.41e5i)3-s − 7.85e7i·5-s − 2.19e9·7-s + (1.65e11 − 2.28e11i)9-s + 1.55e12i·11-s − 3.72e13·13-s + (1.89e13 + 3.71e13i)15-s − 6.73e14i·17-s − 9.37e14·19-s + (1.04e15 − 5.31e14i)21-s + 3.13e16i·23-s + 5.34e16·25-s + (−2.30e16 + 1.48e17i)27-s − 2.21e17i·29-s − 6.76e17·31-s + ⋯ |
| L(s) = 1 | + (−0.890 + 0.454i)3-s − 0.321i·5-s − 0.158·7-s + (0.586 − 0.810i)9-s + 0.496i·11-s − 1.59·13-s + (0.146 + 0.286i)15-s − 1.15i·17-s − 0.423·19-s + (0.141 − 0.0722i)21-s + 1.43i·23-s + 0.896·25-s + (−0.153 + 0.988i)27-s − 0.625i·29-s − 0.858·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{25}{2})\) |
\(\approx\) |
\(0.7003215309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7003215309\) |
| \(L(13)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.73e5 - 2.41e5i)T \) |
| good | 5 | \( 1 + 7.85e7iT - 5.96e16T^{2} \) |
| 7 | \( 1 + 2.19e9T + 1.91e20T^{2} \) |
| 11 | \( 1 - 1.55e12iT - 9.84e24T^{2} \) |
| 13 | \( 1 + 3.72e13T + 5.42e26T^{2} \) |
| 17 | \( 1 + 6.73e14iT - 3.39e29T^{2} \) |
| 19 | \( 1 + 9.37e14T + 4.89e30T^{2} \) |
| 23 | \( 1 - 3.13e16iT - 4.80e32T^{2} \) |
| 29 | \( 1 + 2.21e17iT - 1.25e35T^{2} \) |
| 31 | \( 1 + 6.76e17T + 6.20e35T^{2} \) |
| 37 | \( 1 + 4.23e18T + 4.33e37T^{2} \) |
| 41 | \( 1 - 2.43e19iT - 5.09e38T^{2} \) |
| 43 | \( 1 + 2.27e19T + 1.59e39T^{2} \) |
| 47 | \( 1 - 6.93e19iT - 1.35e40T^{2} \) |
| 53 | \( 1 + 8.43e20iT - 2.41e41T^{2} \) |
| 59 | \( 1 + 2.79e21iT - 3.16e42T^{2} \) |
| 61 | \( 1 + 4.56e21T + 7.04e42T^{2} \) |
| 67 | \( 1 - 1.20e21T + 6.69e43T^{2} \) |
| 71 | \( 1 + 1.60e22iT - 2.69e44T^{2} \) |
| 73 | \( 1 - 2.60e22T + 5.24e44T^{2} \) |
| 79 | \( 1 + 8.00e22T + 3.49e45T^{2} \) |
| 83 | \( 1 + 1.65e23iT - 1.14e46T^{2} \) |
| 89 | \( 1 - 4.33e23iT - 6.10e46T^{2} \) |
| 97 | \( 1 + 4.08e23T + 4.81e47T^{2} \) |
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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
−11.24489710175000111725832205219, −9.900501726323055781373215923637, −9.359733945751488562423152160401, −7.58026681581507660439109675548, −6.62504903906586747382830510794, −5.19271526293877761350066269860, −4.69463287374988064756440512828, −3.24385054791430515753068948248, −1.78663963224908624552469841819, −0.41448621220614708679735584858,
0.33258003434375767280224820878, 1.66024677699621693779982042617, 2.77501795465280630056773067351, 4.34147617981676823467945739496, 5.43082127189944882613186733906, 6.52300277198292783501870108886, 7.34905457657965594441626238299, 8.671449932547360145754123684853, 10.25951307009348058937846413287, 10.87079507611043848223488066658
