| L(s) = 1 | − 3-s − 4.32·5-s + 1.39·7-s + 9-s − 1.39·11-s + 1.72·13-s + 4.32·15-s + 0.601·17-s + 19-s − 1.39·21-s − 8.36·23-s + 13.6·25-s − 27-s − 4.79·29-s + 9.57·31-s + 1.39·33-s − 6.04·35-s + 1.72·37-s − 1.72·39-s + 6.64·41-s + 10.0·43-s − 4.32·45-s − 9.76·47-s − 5.04·49-s − 0.601·51-s + 7.29·53-s + 6.04·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.93·5-s + 0.528·7-s + 0.333·9-s − 0.421·11-s + 0.477·13-s + 1.11·15-s + 0.145·17-s + 0.229·19-s − 0.305·21-s − 1.74·23-s + 2.73·25-s − 0.192·27-s − 0.890·29-s + 1.71·31-s + 0.243·33-s − 1.02·35-s + 0.283·37-s − 0.275·39-s + 1.03·41-s + 1.53·43-s − 0.644·45-s − 1.42·47-s − 0.720·49-s − 0.0842·51-s + 1.00·53-s + 0.815·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 4.32T + 5T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 17 | \( 1 - 0.601T + 17T^{2} \) |
| 23 | \( 1 + 8.36T + 23T^{2} \) |
| 29 | \( 1 + 4.79T + 29T^{2} \) |
| 31 | \( 1 - 9.57T + 31T^{2} \) |
| 37 | \( 1 - 1.72T + 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 9.76T + 47T^{2} \) |
| 53 | \( 1 - 7.29T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 9.24T + 61T^{2} \) |
| 67 | \( 1 + 0.646T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 2.12T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 3.59T + 89T^{2} \) |
| 97 | \( 1 + 5.05T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.896707331601161308301135019561, −7.74676700367980843910487464539, −6.76847486566383764603942071469, −5.92497267741604382148347706470, −4.99880785994907739684088646975, −4.22190161198305047995717360786, −3.77521679107942566980371145119, −2.62135208935793011816959748481, −1.11030680139960527543608185608, 0,
1.11030680139960527543608185608, 2.62135208935793011816959748481, 3.77521679107942566980371145119, 4.22190161198305047995717360786, 4.99880785994907739684088646975, 5.92497267741604382148347706470, 6.76847486566383764603942071469, 7.74676700367980843910487464539, 7.896707331601161308301135019561
