| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 13-s − 14-s + 16-s − 6·17-s − 4·19-s − 6·23-s − 5·25-s − 26-s + 28-s + 8·31-s − 32-s + 6·34-s + 2·37-s + 4·38-s − 4·43-s + 6·46-s − 6·47-s + 49-s + 5·50-s + 52-s − 56-s − 6·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 1.25·23-s − 25-s − 0.196·26-s + 0.188·28-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.648·38-s − 0.609·43-s + 0.884·46-s − 0.875·47-s + 1/7·49-s + 0.707·50-s + 0.138·52-s − 0.133·56-s − 0.781·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.833123066969410272291086657384, −8.315380262939132711743531410580, −7.59113649640036618050579447442, −6.52107659919036530841891307507, −6.05514828155563894984871537986, −4.73802182050113221543979801436, −3.94161510269754508782544806387, −2.55291695292451487866541530230, −1.66895190682184687500782568597, 0,
1.66895190682184687500782568597, 2.55291695292451487866541530230, 3.94161510269754508782544806387, 4.73802182050113221543979801436, 6.05514828155563894984871537986, 6.52107659919036530841891307507, 7.59113649640036618050579447442, 8.315380262939132711743531410580, 8.833123066969410272291086657384
