| L(s) = 1 | − 3-s − 3·7-s + 9-s − 3·11-s − 3·13-s + 17-s − 4·19-s + 3·21-s − 2·23-s − 5·25-s − 27-s + 29-s − 2·31-s + 3·33-s − 6·37-s + 3·39-s + 10·41-s − 3·47-s + 2·49-s − 51-s + 4·53-s + 4·57-s + 10·59-s − 6·61-s − 3·63-s + 3·67-s + 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.832·13-s + 0.242·17-s − 0.917·19-s + 0.654·21-s − 0.417·23-s − 25-s − 0.192·27-s + 0.185·29-s − 0.359·31-s + 0.522·33-s − 0.986·37-s + 0.480·39-s + 1.56·41-s − 0.437·47-s + 2/7·49-s − 0.140·51-s + 0.549·53-s + 0.529·57-s + 1.30·59-s − 0.768·61-s − 0.377·63-s + 0.366·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−10.93467832649205170721115272938, −10.12008600907709017461501290275, −9.456675890582331744607437581213, −8.111098069009176049402214600428, −7.09826106540540756744015299424, −6.14164787642859827596032120545, −5.19861359947890845670001910116, −3.87599905929010573440260747091, −2.43549171570191583756755894426, 0,
2.43549171570191583756755894426, 3.87599905929010573440260747091, 5.19861359947890845670001910116, 6.14164787642859827596032120545, 7.09826106540540756744015299424, 8.111098069009176049402214600428, 9.456675890582331744607437581213, 10.12008600907709017461501290275, 10.93467832649205170721115272938
