| L(s) = 1 | + 4.70·2-s + 3·3-s + 14.1·4-s − 15.5·5-s + 14.1·6-s + 28.9·8-s + 9·9-s − 72.9·10-s − 11·11-s + 42.4·12-s + 67.8·13-s − 46.5·15-s + 23.1·16-s − 132.·17-s + 42.3·18-s − 44.2·19-s − 219.·20-s − 51.7·22-s + 44.9·23-s + 86.9·24-s + 115.·25-s + 319.·26-s + 27·27-s − 99.9·29-s − 218.·30-s − 275.·31-s − 122.·32-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.577·3-s + 1.76·4-s − 1.38·5-s + 0.960·6-s + 1.28·8-s + 0.333·9-s − 2.30·10-s − 0.301·11-s + 1.02·12-s + 1.44·13-s − 0.800·15-s + 0.361·16-s − 1.88·17-s + 0.554·18-s − 0.533·19-s − 2.45·20-s − 0.501·22-s + 0.407·23-s + 0.739·24-s + 0.924·25-s + 2.40·26-s + 0.192·27-s − 0.640·29-s − 1.33·30-s − 1.59·31-s − 0.679·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 4.70T + 8T^{2} \) |
| 5 | \( 1 + 15.5T + 125T^{2} \) |
| 13 | \( 1 - 67.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 132.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 44.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 99.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 275.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 438.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 233.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 294.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 247.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 406.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 201.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 400.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 342.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 602.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 254.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 604.T + 9.12e5T^{2} \) |
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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
−8.507834384144643420332156877891, −7.68218543836875494715161914881, −6.83213810648068092397664181051, −6.18358781526411296298748274396, −5.01860617233980386102849467104, −4.18167519402746165499638820492, −3.77445946280739761729989166674, −2.93586949099991550846309199612, −1.82674360579125943933669988774, 0,
1.82674360579125943933669988774, 2.93586949099991550846309199612, 3.77445946280739761729989166674, 4.18167519402746165499638820492, 5.01860617233980386102849467104, 6.18358781526411296298748274396, 6.83213810648068092397664181051, 7.68218543836875494715161914881, 8.507834384144643420332156877891
