| L(s) = 1 | − 4·7-s − 13-s + 17-s − 4·23-s + 6·29-s − 4·31-s + 2·37-s + 6·41-s + 6·43-s − 12·47-s + 9·49-s + 8·53-s − 10·59-s − 8·61-s − 8·67-s − 8·73-s + 2·79-s + 4·83-s − 4·89-s + 4·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 4·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.277·13-s + 0.242·17-s − 0.834·23-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.937·41-s + 0.914·43-s − 1.75·47-s + 9/7·49-s + 1.09·53-s − 1.30·59-s − 1.02·61-s − 0.977·67-s − 0.936·73-s + 0.225·79-s + 0.439·83-s − 0.423·89-s + 0.419·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.366·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5694471306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5694471306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.45642432725445, −12.00252520247965, −11.76627973477411, −10.91625562076532, −10.62813424217638, −10.15013724381797, −9.702978613771675, −9.323801013636083, −9.001385722405054, −8.325197354828716, −7.838415942021594, −7.405586247199631, −6.884486596114897, −6.390051564314765, −6.027328692485801, −5.649232558435270, −4.933566710970275, −4.397889244459871, −3.913996290276856, −3.358911436592445, −2.856312743327818, −2.496740957000455, −1.690805646390647, −1.028231392549004, −0.2092259478384990,
0.2092259478384990, 1.028231392549004, 1.690805646390647, 2.496740957000455, 2.856312743327818, 3.358911436592445, 3.913996290276856, 4.397889244459871, 4.933566710970275, 5.649232558435270, 6.027328692485801, 6.390051564314765, 6.884486596114897, 7.405586247199631, 7.838415942021594, 8.325197354828716, 9.001385722405054, 9.323801013636083, 9.702978613771675, 10.15013724381797, 10.62813424217638, 10.91625562076532, 11.76627973477411, 12.00252520247965, 12.45642432725445
