| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s − 8·6-s + 30·7-s + 8·8-s − 11·9-s + 11·11-s − 16·12-s − 16·13-s + 60·14-s + 16·16-s + 112·17-s − 22·18-s − 64·19-s − 120·21-s + 22·22-s − 36·23-s − 32·24-s − 32·26-s + 152·27-s + 120·28-s + 10·29-s − 48·31-s + 32·32-s − 44·33-s + 224·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 1/2·4-s − 0.544·6-s + 1.61·7-s + 0.353·8-s − 0.407·9-s + 0.301·11-s − 0.384·12-s − 0.341·13-s + 1.14·14-s + 1/4·16-s + 1.59·17-s − 0.288·18-s − 0.772·19-s − 1.24·21-s + 0.213·22-s − 0.326·23-s − 0.272·24-s − 0.241·26-s + 1.08·27-s + 0.809·28-s + 0.0640·29-s − 0.278·31-s + 0.176·32-s − 0.232·33-s + 1.12·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.858075369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.858075369\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 - 112 T + p^{3} T^{2} \) |
| 19 | \( 1 + 64 T + p^{3} T^{2} \) |
| 23 | \( 1 + 36 T + p^{3} T^{2} \) |
| 29 | \( 1 - 10 T + p^{3} T^{2} \) |
| 31 | \( 1 + 48 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 278 T + p^{3} T^{2} \) |
| 43 | \( 1 - 330 T + p^{3} T^{2} \) |
| 47 | \( 1 + 476 T + p^{3} T^{2} \) |
| 53 | \( 1 + 150 T + p^{3} T^{2} \) |
| 59 | \( 1 - 732 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 - 848 T + p^{3} T^{2} \) |
| 71 | \( 1 - 240 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1128 T + p^{3} T^{2} \) |
| 79 | \( 1 - 788 T + p^{3} T^{2} \) |
| 83 | \( 1 - 698 T + p^{3} T^{2} \) |
| 89 | \( 1 + 458 T + p^{3} T^{2} \) |
| 97 | \( 1 + 134 T + p^{3} T^{2} \) |
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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
−10.86627286090103014427330311811, −9.696777041137299453560277951630, −8.310697847728474823777363502839, −7.71365287666300187666777190061, −6.47932446664538653798388319093, −5.51633108715946113189927130719, −4.95848980880920201589228902723, −3.88450774723349078191288293151, −2.34915972207009255896435063607, −1.02028307872739026436833186699,
1.02028307872739026436833186699, 2.34915972207009255896435063607, 3.88450774723349078191288293151, 4.95848980880920201589228902723, 5.51633108715946113189927130719, 6.47932446664538653798388319093, 7.71365287666300187666777190061, 8.310697847728474823777363502839, 9.696777041137299453560277951630, 10.86627286090103014427330311811
