L(s) = 1 | − 1.32·2-s − 3-s − 0.239·4-s + 2.32·5-s + 1.32·6-s − 7-s + 2.97·8-s + 9-s − 3.08·10-s + 0.239·12-s − 4.53·13-s + 1.32·14-s − 2.32·15-s − 3.46·16-s + 3.29·17-s − 1.32·18-s − 1.08·19-s − 0.556·20-s + 21-s + 6.29·23-s − 2.97·24-s + 0.414·25-s + 6.02·26-s − 27-s + 0.239·28-s − 3.16·29-s + 3.08·30-s + ⋯ |
L(s) = 1 | − 0.938·2-s − 0.577·3-s − 0.119·4-s + 1.04·5-s + 0.541·6-s − 0.377·7-s + 1.05·8-s + 0.333·9-s − 0.976·10-s + 0.0690·12-s − 1.25·13-s + 0.354·14-s − 0.600·15-s − 0.866·16-s + 0.799·17-s − 0.312·18-s − 0.249·19-s − 0.124·20-s + 0.218·21-s + 1.31·23-s − 0.606·24-s + 0.0829·25-s + 1.18·26-s − 0.192·27-s + 0.0452·28-s − 0.587·29-s + 0.563·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 + 1.08T + 19T^{2} \) |
| 23 | \( 1 - 6.29T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 + 4.90T + 41T^{2} \) |
| 43 | \( 1 + 8.38T + 43T^{2} \) |
| 47 | \( 1 + 4.69T + 47T^{2} \) |
| 53 | \( 1 + 4.98T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 1.74T + 61T^{2} \) |
| 67 | \( 1 + 9.44T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 0.205T + 83T^{2} \) |
| 89 | \( 1 - 3.93T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.716557468130602997186088778412, −7.75941652114083868750768281471, −7.07799956630823125937980943075, −6.33058993542588461810751723885, −5.24739568346382758616655174464, −4.94443302056571626113482123458, −3.59523360488997679615551829242, −2.29849302183988289665679308363, −1.30862101707854891910608183945, 0,
1.30862101707854891910608183945, 2.29849302183988289665679308363, 3.59523360488997679615551829242, 4.94443302056571626113482123458, 5.24739568346382758616655174464, 6.33058993542588461810751723885, 7.07799956630823125937980943075, 7.75941652114083868750768281471, 8.716557468130602997186088778412