| L(s) = 1 | + 2·2-s + 4·4-s + 6·5-s − 18·7-s + 8·8-s + 12·10-s + 42·13-s − 36·14-s + 16·16-s − 30·17-s + 60·19-s + 24·20-s − 186·23-s − 89·25-s + 84·26-s − 72·28-s − 42·29-s + 16·31-s + 32·32-s − 60·34-s − 108·35-s − 74·37-s + 120·38-s + 48·40-s − 150·41-s − 264·43-s − 372·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.536·5-s − 0.971·7-s + 0.353·8-s + 0.379·10-s + 0.896·13-s − 0.687·14-s + 1/4·16-s − 0.428·17-s + 0.724·19-s + 0.268·20-s − 1.68·23-s − 0.711·25-s + 0.633·26-s − 0.485·28-s − 0.268·29-s + 0.0926·31-s + 0.176·32-s − 0.302·34-s − 0.521·35-s − 0.328·37-s + 0.512·38-s + 0.189·40-s − 0.571·41-s − 0.936·43-s − 1.19·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 60 T + p^{3} T^{2} \) |
| 23 | \( 1 + 186 T + p^{3} T^{2} \) |
| 29 | \( 1 + 42 T + p^{3} T^{2} \) |
| 31 | \( 1 - 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 150 T + p^{3} T^{2} \) |
| 43 | \( 1 + 264 T + p^{3} T^{2} \) |
| 47 | \( 1 - 306 T + p^{3} T^{2} \) |
| 53 | \( 1 - 126 T + p^{3} T^{2} \) |
| 59 | \( 1 - 492 T + p^{3} T^{2} \) |
| 61 | \( 1 - 6 T + p^{3} T^{2} \) |
| 67 | \( 1 + 524 T + p^{3} T^{2} \) |
| 71 | \( 1 - 390 T + p^{3} T^{2} \) |
| 73 | \( 1 + 984 T + p^{3} T^{2} \) |
| 79 | \( 1 - 354 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1116 T + p^{3} T^{2} \) |
| 89 | \( 1 - 744 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1406 T + p^{3} T^{2} \) |
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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
−8.304023975282634185033341706569, −7.37584474431183811900174260305, −6.47643603109655477850886148282, −5.99782411241349773134535576432, −5.28260034163657000580835181350, −4.07634626691094693693940184739, −3.48995842869761710384214519000, −2.46257243649055617448881277253, −1.48619269560460824384265926658, 0,
1.48619269560460824384265926658, 2.46257243649055617448881277253, 3.48995842869761710384214519000, 4.07634626691094693693940184739, 5.28260034163657000580835181350, 5.99782411241349773134535576432, 6.47643603109655477850886148282, 7.37584474431183811900174260305, 8.304023975282634185033341706569
