| L(s) = 1 | − 3-s − 2.66·5-s − 7-s + 9-s − 1.57·11-s + 13-s + 2.66·15-s + 4.75·17-s + 2.23·19-s + 21-s − 5.84·23-s + 2.08·25-s − 27-s + 4.23·29-s − 7.28·31-s + 1.57·33-s + 2.66·35-s + 10.4·37-s − 39-s − 2.25·41-s − 0.913·43-s − 2.66·45-s + 2.09·47-s + 49-s − 4.75·51-s + 1.08·53-s + 4.19·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.19·5-s − 0.377·7-s + 0.333·9-s − 0.475·11-s + 0.277·13-s + 0.687·15-s + 1.15·17-s + 0.513·19-s + 0.218·21-s − 1.21·23-s + 0.417·25-s − 0.192·27-s + 0.786·29-s − 1.30·31-s + 0.274·33-s + 0.449·35-s + 1.72·37-s − 0.160·39-s − 0.351·41-s − 0.139·43-s − 0.396·45-s + 0.305·47-s + 0.142·49-s − 0.666·51-s + 0.149·53-s + 0.565·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.66T + 5T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 + 7.28T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 2.25T + 41T^{2} \) |
| 43 | \( 1 + 0.913T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 - 1.08T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 7.51T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 7.42T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.891101562585154362139018717759, −7.41507359338878937961642364443, −6.55572887951952351357695210457, −5.74003334418831623223860166038, −5.11452076224163019924266588334, −4.04804755671374475120997536793, −3.62096671539881266056804701561, −2.55137885659551782993702748081, −1.09618222228326484542116341695, 0,
1.09618222228326484542116341695, 2.55137885659551782993702748081, 3.62096671539881266056804701561, 4.04804755671374475120997536793, 5.11452076224163019924266588334, 5.74003334418831623223860166038, 6.55572887951952351357695210457, 7.41507359338878937961642364443, 7.891101562585154362139018717759
