| L(s) = 1 | − 3-s + 9-s − 2·11-s − 3·13-s + 6·17-s − 7·19-s + 6·23-s − 27-s − 2·29-s − 5·31-s + 2·33-s − 10·37-s + 3·39-s − 12·41-s + 3·43-s + 10·47-s − 6·51-s + 7·57-s − 6·59-s + 13·61-s + 7·67-s − 6·69-s + 4·71-s − 6·73-s + 8·79-s + 81-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.832·13-s + 1.45·17-s − 1.60·19-s + 1.25·23-s − 0.192·27-s − 0.371·29-s − 0.898·31-s + 0.348·33-s − 1.64·37-s + 0.480·39-s − 1.87·41-s + 0.457·43-s + 1.45·47-s − 0.840·51-s + 0.927·57-s − 0.781·59-s + 1.66·61-s + 0.855·67-s − 0.722·69-s + 0.474·71-s − 0.702·73-s + 0.900·79-s + 1/9·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8586709825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8586709825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.35170381517690, −13.88097629014715, −13.12978511879702, −12.77630011103087, −12.29069776594771, −11.96008286391128, −11.19601704991993, −10.70200296246878, −10.36261752356937, −9.855741764392457, −9.214483715415679, −8.622117202749960, −8.110871142295001, −7.397967331625229, −7.034671658481523, −6.504315009769867, −5.692849737461520, −5.208415470233721, −4.976859830752952, −4.019549171708897, −3.535579459537731, −2.727861503801789, −2.067712525145013, −1.310086066588166, −0.3354045531350874,
0.3354045531350874, 1.310086066588166, 2.067712525145013, 2.727861503801789, 3.535579459537731, 4.019549171708897, 4.976859830752952, 5.208415470233721, 5.692849737461520, 6.504315009769867, 7.034671658481523, 7.397967331625229, 8.110871142295001, 8.622117202749960, 9.214483715415679, 9.855741764392457, 10.36261752356937, 10.70200296246878, 11.19601704991993, 11.96008286391128, 12.29069776594771, 12.77630011103087, 13.12978511879702, 13.88097629014715, 14.35170381517690
