| L(s) = 1 | + 2·5-s − 7·7-s + 12·11-s − 66·13-s + 70·17-s + 92·19-s + 16·23-s − 121·25-s + 122·29-s − 64·31-s − 14·35-s − 306·37-s − 50·41-s − 20·43-s − 176·47-s + 49·49-s − 526·53-s + 24·55-s + 540·59-s − 818·61-s − 132·65-s + 228·67-s + 864·71-s + 106·73-s − 84·77-s − 736·79-s − 588·83-s + ⋯ |
| L(s) = 1 | + 0.178·5-s − 0.377·7-s + 0.328·11-s − 1.40·13-s + 0.998·17-s + 1.11·19-s + 0.145·23-s − 0.967·25-s + 0.781·29-s − 0.370·31-s − 0.0676·35-s − 1.35·37-s − 0.190·41-s − 0.0709·43-s − 0.546·47-s + 1/7·49-s − 1.36·53-s + 0.0588·55-s + 1.19·59-s − 1.71·61-s − 0.251·65-s + 0.415·67-s + 1.44·71-s + 0.169·73-s − 0.124·77-s − 1.04·79-s − 0.777·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 66 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 16 T + p^{3} T^{2} \) |
| 29 | \( 1 - 122 T + p^{3} T^{2} \) |
| 31 | \( 1 + 64 T + p^{3} T^{2} \) |
| 37 | \( 1 + 306 T + p^{3} T^{2} \) |
| 41 | \( 1 + 50 T + p^{3} T^{2} \) |
| 43 | \( 1 + 20 T + p^{3} T^{2} \) |
| 47 | \( 1 + 176 T + p^{3} T^{2} \) |
| 53 | \( 1 + 526 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 + 818 T + p^{3} T^{2} \) |
| 67 | \( 1 - 228 T + p^{3} T^{2} \) |
| 71 | \( 1 - 864 T + p^{3} T^{2} \) |
| 73 | \( 1 - 106 T + p^{3} T^{2} \) |
| 79 | \( 1 + 736 T + p^{3} T^{2} \) |
| 83 | \( 1 + 588 T + p^{3} T^{2} \) |
| 89 | \( 1 + 146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1214 T + p^{3} T^{2} \) |
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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
−9.527887713264085006153648823952, −8.262931769354417745287759580429, −7.44890689331092643028532519413, −6.71626769113060529874415555158, −5.61490509335485506162266311824, −4.91849508292469879399364407742, −3.65792627424099809792997676115, −2.74313345203200880135741286713, −1.44409587351650044580748067482, 0,
1.44409587351650044580748067482, 2.74313345203200880135741286713, 3.65792627424099809792997676115, 4.91849508292469879399364407742, 5.61490509335485506162266311824, 6.71626769113060529874415555158, 7.44890689331092643028532519413, 8.262931769354417745287759580429, 9.527887713264085006153648823952
