| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 5·5-s − 6·6-s + 8·8-s + 9·9-s − 10·10-s − 15·11-s − 12·12-s + 77·13-s + 15·15-s + 16·16-s − 96·17-s + 18·18-s − 37·19-s − 20·20-s − 30·22-s − 99·23-s − 24·24-s + 25·25-s + 154·26-s − 27·27-s + 240·29-s + 30·30-s − 166·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.411·11-s − 0.288·12-s + 1.64·13-s + 0.258·15-s + 1/4·16-s − 1.36·17-s + 0.235·18-s − 0.446·19-s − 0.223·20-s − 0.290·22-s − 0.897·23-s − 0.204·24-s + 1/5·25-s + 1.16·26-s − 0.192·27-s + 1.53·29-s + 0.182·30-s − 0.961·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 - 77 T + p^{3} T^{2} \) |
| 17 | \( 1 + 96 T + p^{3} T^{2} \) |
| 19 | \( 1 + 37 T + p^{3} T^{2} \) |
| 23 | \( 1 + 99 T + p^{3} T^{2} \) |
| 29 | \( 1 - 240 T + p^{3} T^{2} \) |
| 31 | \( 1 + 166 T + p^{3} T^{2} \) |
| 37 | \( 1 - 335 T + p^{3} T^{2} \) |
| 41 | \( 1 - 21 T + p^{3} T^{2} \) |
| 43 | \( 1 + 40 T + p^{3} T^{2} \) |
| 47 | \( 1 + 639 T + p^{3} T^{2} \) |
| 53 | \( 1 - 153 T + p^{3} T^{2} \) |
| 59 | \( 1 + 684 T + p^{3} T^{2} \) |
| 61 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 608 T + p^{3} T^{2} \) |
| 71 | \( 1 - 198 T + p^{3} T^{2} \) |
| 73 | \( 1 - 338 T + p^{3} T^{2} \) |
| 79 | \( 1 + 736 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + 1290 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1456 T + p^{3} T^{2} \) |
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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
−8.504079750583663477982502246362, −7.981742839785028439158230978423, −6.70398069022327413833055690495, −6.35084233824843406277561019571, −5.39660556024295706807496639092, −4.41328387586736677945791163350, −3.84047628773428451780117102662, −2.63110459849254670526067483010, −1.38172208924909856523945126279, 0,
1.38172208924909856523945126279, 2.63110459849254670526067483010, 3.84047628773428451780117102662, 4.41328387586736677945791163350, 5.39660556024295706807496639092, 6.35084233824843406277561019571, 6.70398069022327413833055690495, 7.981742839785028439158230978423, 8.504079750583663477982502246362
