L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s − 4·11-s + 4·13-s − 2·15-s + 4·17-s − 6·19-s + 21-s − 23-s − 25-s − 27-s + 2·29-s + 2·31-s + 4·33-s − 2·35-s − 2·37-s − 4·39-s − 10·41-s + 8·43-s + 2·45-s + 6·47-s + 49-s − 4·51-s − 6·53-s − 8·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s − 0.516·15-s + 0.970·17-s − 1.37·19-s + 0.218·21-s − 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.696·33-s − 0.338·35-s − 0.328·37-s − 0.640·39-s − 1.56·41-s + 1.21·43-s + 0.298·45-s + 0.875·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−7.47862845048392761983528177667, −6.62276328439378980118231401299, −5.97967124637393041588056885661, −5.66249953835192057095879727766, −4.82816813990084514453959126499, −3.96986064994753449176908443473, −3.05349248317122730564165195580, −2.18510118234879064977259634115, −1.27058822845499447423434912866, 0,
1.27058822845499447423434912866, 2.18510118234879064977259634115, 3.05349248317122730564165195580, 3.96986064994753449176908443473, 4.82816813990084514453959126499, 5.66249953835192057095879727766, 5.97967124637393041588056885661, 6.62276328439378980118231401299, 7.47862845048392761983528177667