| L(s) = 1 | + 2.10·2-s − 3·3-s − 3.55·4-s + 5.76·5-s − 6.32·6-s − 24.3·8-s + 9·9-s + 12.1·10-s + 11·11-s + 10.6·12-s + 85.4·13-s − 17.3·15-s − 22.8·16-s − 110.·17-s + 18.9·18-s + 4.28·19-s − 20.5·20-s + 23.1·22-s − 147.·23-s + 73.0·24-s − 91.7·25-s + 180.·26-s − 27·27-s + 147.·29-s − 36.4·30-s − 118.·31-s + 146.·32-s + ⋯ |
| L(s) = 1 | + 0.745·2-s − 0.577·3-s − 0.444·4-s + 0.515·5-s − 0.430·6-s − 1.07·8-s + 0.333·9-s + 0.384·10-s + 0.301·11-s + 0.256·12-s + 1.82·13-s − 0.297·15-s − 0.357·16-s − 1.57·17-s + 0.248·18-s + 0.0517·19-s − 0.229·20-s + 0.224·22-s − 1.34·23-s + 0.621·24-s − 0.733·25-s + 1.35·26-s − 0.192·27-s + 0.946·29-s − 0.221·30-s − 0.688·31-s + 0.810·32-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 | \( 1 - 2.10T + 8T^{2} \) |
| 5 | \( 1 - 5.76T + 125T^{2} \) |
| 13 | \( 1 - 85.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.28T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 64.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 489.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 429.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 286.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 114.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 647.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 611.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 565.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 268.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 740.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 408.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 672.T + 9.12e5T^{2} \) |
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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
−8.825582400382017611857416086765, −7.87475814868479852527832203655, −6.49446681207473854089951153520, −6.10770570438888416226954657488, −5.47144979799697073951980847698, −4.20244057990904554165992937609, −4.01423570331931245366244440277, −2.56575636418331924850364165842, −1.29545570985379190111442490000, 0,
1.29545570985379190111442490000, 2.56575636418331924850364165842, 4.01423570331931245366244440277, 4.20244057990904554165992937609, 5.47144979799697073951980847698, 6.10770570438888416226954657488, 6.49446681207473854089951153520, 7.87475814868479852527832203655, 8.825582400382017611857416086765
