L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 9-s + 11-s + 4·15-s + 4·17-s − 4·19-s − 2·21-s − 4·23-s − 25-s + 4·27-s − 2·29-s − 2·31-s − 2·33-s − 2·35-s + 6·37-s + 4·41-s + 4·43-s − 2·45-s + 2·47-s + 49-s − 8·51-s − 2·53-s − 2·55-s + 8·57-s + 6·59-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.03·15-s + 0.970·17-s − 0.917·19-s − 0.436·21-s − 0.834·23-s − 1/5·25-s + 0.769·27-s − 0.371·29-s − 0.359·31-s − 0.348·33-s − 0.338·35-s + 0.986·37-s + 0.624·41-s + 0.609·43-s − 0.298·45-s + 0.291·47-s + 1/7·49-s − 1.12·51-s − 0.274·53-s − 0.269·55-s + 1.05·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.81471559919429316687239101837, −7.24015489942259604405440552497, −6.22474682847289765271955059501, −5.84573150094526730078225656213, −4.95890967095362737510025794037, −4.23076734373564530341685680553, −3.57643311630839300149423314898, −2.31814487044434135855216314443, −1.05262565113522034371033309180, 0,
1.05262565113522034371033309180, 2.31814487044434135855216314443, 3.57643311630839300149423314898, 4.23076734373564530341685680553, 4.95890967095362737510025794037, 5.84573150094526730078225656213, 6.22474682847289765271955059501, 7.24015489942259604405440552497, 7.81471559919429316687239101837