| L(s) = 1 | − 3-s + 3.90i·5-s + 0.0794i·7-s + 9-s − 1.50i·11-s − 3.90i·15-s + 6.52·17-s − 0.786i·19-s − 0.0794i·21-s + 7.03·23-s − 10.2·25-s − 27-s + 6.15·29-s + 1.43i·31-s + 1.50i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.74i·5-s + 0.0300i·7-s + 0.333·9-s − 0.453i·11-s − 1.00i·15-s + 1.58·17-s − 0.180i·19-s − 0.0173i·21-s + 1.46·23-s − 2.05·25-s − 0.192·27-s + 1.14·29-s + 0.258i·31-s + 0.261i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.727425642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727425642\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.90iT - 5T^{2} \) |
| 7 | \( 1 - 0.0794iT - 7T^{2} \) |
| 11 | \( 1 + 1.50iT - 11T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 + 0.786iT - 19T^{2} \) |
| 23 | \( 1 - 7.03T + 23T^{2} \) |
| 29 | \( 1 - 6.15T + 29T^{2} \) |
| 31 | \( 1 - 1.43iT - 31T^{2} \) |
| 37 | \( 1 + 9.23iT - 37T^{2} \) |
| 41 | \( 1 - 7.75iT - 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 + 12.7iT - 47T^{2} \) |
| 53 | \( 1 - 2.73T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 7.80iT - 67T^{2} \) |
| 71 | \( 1 + 15.1iT - 71T^{2} \) |
| 73 | \( 1 + 4.16iT - 73T^{2} \) |
| 79 | \( 1 - 3.83T + 79T^{2} \) |
| 83 | \( 1 - 9.04iT - 83T^{2} \) |
| 89 | \( 1 - 4.75iT - 89T^{2} \) |
| 97 | \( 1 - 6.94iT - 97T^{2} \) |
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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
−8.379966095006169102552677946460, −7.58371712993194982876958265882, −6.96144464617281267289654663948, −6.43521087764510522512741051982, −5.65762655613969240876045297823, −4.95465285944380665890939533521, −3.64125061135183114686584922677, −3.18888819300109599037449055633, −2.24440854864271795156865907579, −0.806837098860914948414707992436,
0.883714965217188065033221087184, 1.35319660302594001234294170461, 2.82854778713578791029379161398, 4.04740662056655869772291782208, 4.69195776126605805367720487141, 5.35341459388270047829108847214, 5.83781820478821975700354176758, 6.94674834887062992067384659124, 7.65153729152865240862899377709, 8.463592530367457989595391597392
