| L(s) = 1 | − 4·5-s − 7-s − 2·11-s − 6·13-s + 4·17-s − 4·19-s − 2·23-s + 11·25-s + 2·29-s + 4·35-s + 2·37-s − 4·43-s − 12·47-s + 49-s + 6·53-s + 8·55-s + 8·59-s + 6·61-s + 24·65-s − 8·67-s − 14·71-s − 2·73-s + 2·77-s + 12·79-s + 4·83-s − 16·85-s + 6·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s − 0.603·11-s − 1.66·13-s + 0.970·17-s − 0.917·19-s − 0.417·23-s + 11/5·25-s + 0.371·29-s + 0.676·35-s + 0.328·37-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s + 1.04·59-s + 0.768·61-s + 2.97·65-s − 0.977·67-s − 1.66·71-s − 0.234·73-s + 0.227·77-s + 1.35·79-s + 0.439·83-s − 1.73·85-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−11.83481765842467689738267568685, −10.65141343664047244023029939050, −9.763849160343204480377812055766, −8.349702149825146102162236283121, −7.69907439436331841244821924582, −6.81660893407171735772005884639, −5.13729108364941390504198607970, −4.08565362497912592132601547351, −2.85749596756875682654277889497, 0,
2.85749596756875682654277889497, 4.08565362497912592132601547351, 5.13729108364941390504198607970, 6.81660893407171735772005884639, 7.69907439436331841244821924582, 8.349702149825146102162236283121, 9.763849160343204480377812055766, 10.65141343664047244023029939050, 11.83481765842467689738267568685
