| L(s) = 1 | − 2·3-s + 7-s + 9-s − 11-s + 2·13-s + 6·17-s + 4·19-s − 2·21-s + 4·23-s − 5·25-s + 4·27-s + 29-s − 4·31-s + 2·33-s − 6·37-s − 4·39-s + 2·41-s − 4·43-s + 49-s − 12·51-s + 2·53-s − 8·57-s + 8·61-s + 63-s − 4·67-s − 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.436·21-s + 0.834·23-s − 25-s + 0.769·27-s + 0.185·29-s − 0.718·31-s + 0.348·33-s − 0.986·37-s − 0.640·39-s + 0.312·41-s − 0.609·43-s + 1/7·49-s − 1.68·51-s + 0.274·53-s − 1.05·57-s + 1.02·61-s + 0.125·63-s − 0.488·67-s − 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| 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
−13.62407104171106, −13.13096512926199, −12.42144012760356, −12.20085537098052, −11.63665117654807, −11.34021583174273, −10.80073735503559, −10.40854823806551, −9.878159580913370, −9.406321998160397, −8.762912385055494, −8.192231986014009, −7.766019314872613, −7.172909534116161, −6.744151479426389, −6.037474988556747, −5.599624675345665, −5.245841603471803, −4.891102749183068, −4.014974536874105, −3.481382116961173, −2.965712327457990, −2.094923178185558, −1.302867871926628, −0.8703612091641825, 0,
0.8703612091641825, 1.302867871926628, 2.094923178185558, 2.965712327457990, 3.481382116961173, 4.014974536874105, 4.891102749183068, 5.245841603471803, 5.599624675345665, 6.037474988556747, 6.744151479426389, 7.172909534116161, 7.766019314872613, 8.192231986014009, 8.762912385055494, 9.406321998160397, 9.878159580913370, 10.40854823806551, 10.80073735503559, 11.34021583174273, 11.63665117654807, 12.20085537098052, 12.42144012760356, 13.13096512926199, 13.62407104171106
