| L(s) = 1 | − 5-s − 4·7-s − 4·11-s + 2·13-s + 2·17-s − 8·23-s + 25-s + 6·29-s + 4·31-s + 4·35-s + 10·37-s − 2·41-s − 12·43-s + 9·49-s + 6·53-s + 4·55-s − 10·61-s − 2·65-s − 4·67-s + 8·71-s + 2·73-s + 16·77-s − 12·79-s − 8·83-s − 2·85-s + 6·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.676·35-s + 1.64·37-s − 0.312·41-s − 1.82·43-s + 9/7·49-s + 0.824·53-s + 0.539·55-s − 1.28·61-s − 0.248·65-s − 0.488·67-s + 0.949·71-s + 0.234·73-s + 1.82·77-s − 1.35·79-s − 0.878·83-s − 0.216·85-s + 0.635·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6812915051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6812915051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.88142105340624, −12.42567575393407, −11.79319886604924, −11.69175997112721, −10.87452730632144, −10.35723091503956, −10.02056360551729, −9.809512894923366, −9.144352589451309, −8.408144802246199, −8.262226391390658, −7.670660048341931, −7.225291494397817, −6.499731794206677, −6.238563639847776, −5.809884990817210, −5.156567121069977, −4.572118236696464, −4.001986297461483, −3.516555118370975, −2.880775789597777, −2.693026471706233, −1.820906919101894, −0.9600740187860227, −0.2596428986599282,
0.2596428986599282, 0.9600740187860227, 1.820906919101894, 2.693026471706233, 2.880775789597777, 3.516555118370975, 4.001986297461483, 4.572118236696464, 5.156567121069977, 5.809884990817210, 6.238563639847776, 6.499731794206677, 7.225291494397817, 7.670660048341931, 8.262226391390658, 8.408144802246199, 9.144352589451309, 9.809512894923366, 10.02056360551729, 10.35723091503956, 10.87452730632144, 11.69175997112721, 11.79319886604924, 12.42567575393407, 12.88142105340624
