| L(s) = 1 | − 2-s + 4-s + 3·7-s − 8-s − 11-s + 13-s − 3·14-s + 16-s + 3·17-s + 3·19-s + 22-s + 23-s − 5·25-s − 26-s + 3·28-s + 4·29-s − 6·31-s − 32-s − 3·34-s − 37-s − 3·38-s + 10·41-s + 12·43-s − 44-s − 46-s + 6·47-s + 2·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.13·7-s − 0.353·8-s − 0.301·11-s + 0.277·13-s − 0.801·14-s + 1/4·16-s + 0.727·17-s + 0.688·19-s + 0.213·22-s + 0.208·23-s − 25-s − 0.196·26-s + 0.566·28-s + 0.742·29-s − 1.07·31-s − 0.176·32-s − 0.514·34-s − 0.164·37-s − 0.486·38-s + 1.56·41-s + 1.82·43-s − 0.150·44-s − 0.147·46-s + 0.875·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.253384391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253384391\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.56855320922795157861129741430, −9.590711301042029947842936857345, −8.792199929808722838659257912612, −7.80630444762192273011758362778, −7.43313124434508075165402720464, −6.01497418650790910506005868954, −5.18717331204885286486593982487, −3.90575639700327756731471854422, −2.46404015546282379837947782123, −1.15779711357483022032503243329,
1.15779711357483022032503243329, 2.46404015546282379837947782123, 3.90575639700327756731471854422, 5.18717331204885286486593982487, 6.01497418650790910506005868954, 7.43313124434508075165402720464, 7.80630444762192273011758362778, 8.792199929808722838659257912612, 9.590711301042029947842936857345, 10.56855320922795157861129741430
