| L(s) = 1 | + 2-s + 1.77·3-s + 4-s + 0.134·5-s + 1.77·6-s + 1.25·7-s + 8-s + 0.145·9-s + 0.134·10-s − 4.38·11-s + 1.77·12-s + 5.80·13-s + 1.25·14-s + 0.239·15-s + 16-s + 3.23·17-s + 0.145·18-s + 5.70·19-s + 0.134·20-s + 2.22·21-s − 4.38·22-s − 23-s + 1.77·24-s − 4.98·25-s + 5.80·26-s − 5.06·27-s + 1.25·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.02·3-s + 0.5·4-s + 0.0603·5-s + 0.724·6-s + 0.473·7-s + 0.353·8-s + 0.0486·9-s + 0.0426·10-s − 1.32·11-s + 0.512·12-s + 1.61·13-s + 0.334·14-s + 0.0618·15-s + 0.250·16-s + 0.785·17-s + 0.0343·18-s + 1.30·19-s + 0.0301·20-s + 0.484·21-s − 0.935·22-s − 0.208·23-s + 0.362·24-s − 0.996·25-s + 1.13·26-s − 0.974·27-s + 0.236·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.846542091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.846542091\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.77T + 3T^{2} \) |
| 5 | \( 1 - 0.134T + 5T^{2} \) |
| 7 | \( 1 - 1.25T + 7T^{2} \) |
| 11 | \( 1 + 4.38T + 11T^{2} \) |
| 13 | \( 1 - 5.80T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 31 | \( 1 - 8.77T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 + 9.04T + 41T^{2} \) |
| 43 | \( 1 - 6.33T + 43T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 53 | \( 1 - 2.72T + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 + 1.88T + 61T^{2} \) |
| 67 | \( 1 + 5.13T + 67T^{2} \) |
| 71 | \( 1 + 9.21T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 2.83T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 0.677T + 97T^{2} \) |
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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
−9.691046867926554540150129389473, −8.516016121178522384735737663703, −8.036788044714305403587624591510, −7.42556714954935411564017861587, −6.03485653345294043778459397773, −5.49895319828071057948259982150, −4.37547218243937884994204803757, −3.33820455709001209436318448536, −2.74847481465466313659064946121, −1.46318891412405983995054358863,
1.46318891412405983995054358863, 2.74847481465466313659064946121, 3.33820455709001209436318448536, 4.37547218243937884994204803757, 5.49895319828071057948259982150, 6.03485653345294043778459397773, 7.42556714954935411564017861587, 8.036788044714305403587624591510, 8.516016121178522384735737663703, 9.691046867926554540150129389473
