| L(s) = 1 | − 5.06·2-s + 17.6·4-s − 27.4·7-s − 48.8·8-s − 11·11-s − 22.6·13-s + 138.·14-s + 106.·16-s − 41.1·17-s − 142.·19-s + 55.7·22-s + 176.·23-s + 114.·26-s − 484.·28-s − 76.2·29-s + 197.·31-s − 147.·32-s + 208.·34-s − 367.·37-s + 719.·38-s + 238.·41-s − 30.2·43-s − 194.·44-s − 892.·46-s + 137.·47-s + 409.·49-s − 400.·52-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 2.20·4-s − 1.48·7-s − 2.16·8-s − 0.301·11-s − 0.484·13-s + 2.65·14-s + 1.66·16-s − 0.587·17-s − 1.71·19-s + 0.539·22-s + 1.59·23-s + 0.867·26-s − 3.26·28-s − 0.487·29-s + 1.14·31-s − 0.816·32-s + 1.05·34-s − 1.63·37-s + 3.07·38-s + 0.907·41-s − 0.107·43-s − 0.665·44-s − 2.85·46-s + 0.426·47-s + 1.19·49-s − 1.06·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 5.06T + 8T^{2} \) |
| 7 | \( 1 + 27.4T + 343T^{2} \) |
| 13 | \( 1 + 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 41.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 367.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 137.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 638.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 103.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 605.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 704.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 782.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 243.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 532.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 85.1T + 9.12e5T^{2} \) |
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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
−8.519149430722291548355996163658, −7.50343965801159145454635650621, −6.67062661064864691663679187752, −6.55237584174357821896284379596, −5.28609715537040016797420987493, −3.92885235106226738242481190212, −2.77305195686313449509931843105, −2.19618074152477204365103634944, −0.78059936038733887518811808610, 0,
0.78059936038733887518811808610, 2.19618074152477204365103634944, 2.77305195686313449509931843105, 3.92885235106226738242481190212, 5.28609715537040016797420987493, 6.55237584174357821896284379596, 6.67062661064864691663679187752, 7.50343965801159145454635650621, 8.519149430722291548355996163658
