| L(s) = 1 | + 1.84·2-s + 0.689·3-s + 1.39·4-s − 1.44·5-s + 1.27·6-s − 1.11·8-s − 2.52·9-s − 2.65·10-s − 0.737·11-s + 0.963·12-s + 3.57·13-s − 0.994·15-s − 4.84·16-s + 2.70·17-s − 4.65·18-s − 8.41·19-s − 2.01·20-s − 1.36·22-s + 2.69·23-s − 0.765·24-s − 2.92·25-s + 6.58·26-s − 3.80·27-s − 8.46·29-s − 1.83·30-s + 5.65·31-s − 6.70·32-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.398·3-s + 0.698·4-s − 0.644·5-s + 0.518·6-s − 0.392·8-s − 0.841·9-s − 0.840·10-s − 0.222·11-s + 0.278·12-s + 0.990·13-s − 0.256·15-s − 1.21·16-s + 0.655·17-s − 1.09·18-s − 1.92·19-s − 0.450·20-s − 0.289·22-s + 0.562·23-s − 0.156·24-s − 0.584·25-s + 1.29·26-s − 0.733·27-s − 1.57·29-s − 0.334·30-s + 1.01·31-s − 1.18·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 1.84T + 2T^{2} \) |
| 3 | \( 1 - 0.689T + 3T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 11 | \( 1 + 0.737T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 + 8.41T + 19T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 + 8.46T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 7.32T + 37T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 + 8.76T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 - 5.66T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 9.10T + 61T^{2} \) |
| 67 | \( 1 - 0.628T + 67T^{2} \) |
| 71 | \( 1 + 6.79T + 71T^{2} \) |
| 73 | \( 1 + 4.90T + 73T^{2} \) |
| 79 | \( 1 + 6.95T + 79T^{2} \) |
| 83 | \( 1 - 9.64T + 83T^{2} \) |
| 89 | \( 1 + 2.07T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.585390910973924606724972269332, −7.85936844127631674128352588231, −6.80683862166769994644805234677, −6.01130488093847693222222683186, −5.41318628352666890871984045570, −4.39580303523326507539077013022, −3.70157679234444762692314550620, −3.10916452084786174619156061864, −2.01851793223719684466461144425, 0,
2.01851793223719684466461144425, 3.10916452084786174619156061864, 3.70157679234444762692314550620, 4.39580303523326507539077013022, 5.41318628352666890871984045570, 6.01130488093847693222222683186, 6.80683862166769994644805234677, 7.85936844127631674128352588231, 8.585390910973924606724972269332
