| L(s) = 1 | + 2-s + 0.0740·3-s + 4-s − 5-s + 0.0740·6-s + 2.79·7-s + 8-s − 2.99·9-s − 10-s + 4.67·11-s + 0.0740·12-s − 3.91·13-s + 2.79·14-s − 0.0740·15-s + 16-s + 0.808·17-s − 2.99·18-s − 5.45·19-s − 20-s + 0.206·21-s + 4.67·22-s − 5.03·23-s + 0.0740·24-s + 25-s − 3.91·26-s − 0.443·27-s + 2.79·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0427·3-s + 0.5·4-s − 0.447·5-s + 0.0302·6-s + 1.05·7-s + 0.353·8-s − 0.998·9-s − 0.316·10-s + 1.41·11-s + 0.0213·12-s − 1.08·13-s + 0.747·14-s − 0.0191·15-s + 0.250·16-s + 0.196·17-s − 0.705·18-s − 1.25·19-s − 0.223·20-s + 0.0451·21-s + 0.997·22-s − 1.04·23-s + 0.0151·24-s + 0.200·25-s − 0.767·26-s − 0.0853·27-s + 0.528·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0740T + 3T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 - 4.67T + 11T^{2} \) |
| 13 | \( 1 + 3.91T + 13T^{2} \) |
| 17 | \( 1 - 0.808T + 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 + 9.90T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 + 0.428T + 37T^{2} \) |
| 41 | \( 1 + 3.64T + 41T^{2} \) |
| 43 | \( 1 - 8.52T + 43T^{2} \) |
| 47 | \( 1 - 3.92T + 47T^{2} \) |
| 53 | \( 1 + 0.714T + 53T^{2} \) |
| 59 | \( 1 - 4.44T + 59T^{2} \) |
| 61 | \( 1 + 8.58T + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 - 1.01T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 7.93T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−7.51969958549801303524950169478, −7.16889983516593103137006254539, −6.03285230870752541588301037852, −5.66684759216929656350754258325, −4.66351633672109288345297829285, −4.14051412217420931456411961229, −3.42695580988806498303591186023, −2.29378282378729730637187605303, −1.64822149398443041155080251148, 0,
1.64822149398443041155080251148, 2.29378282378729730637187605303, 3.42695580988806498303591186023, 4.14051412217420931456411961229, 4.66351633672109288345297829285, 5.66684759216929656350754258325, 6.03285230870752541588301037852, 7.16889983516593103137006254539, 7.51969958549801303524950169478
