| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s + 2·13-s − 2·15-s + 6·17-s − 4·19-s + 21-s + 4·23-s − 25-s + 27-s − 6·29-s + 8·31-s − 2·35-s + 10·37-s + 2·39-s − 10·41-s + 12·43-s − 2·45-s + 8·47-s + 49-s + 6·51-s − 6·53-s − 4·57-s + 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s − 0.917·19-s + 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.338·35-s + 1.64·37-s + 0.320·39-s − 1.56·41-s + 1.82·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.922005174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922005174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.532192059495699965382007757015, −8.645485390247991723199556988296, −7.972516854397409118504829535035, −7.47763526176816074486193675448, −6.39849465748652448924567504195, −5.36249482889116442170023690841, −4.25172242842989871526133469504, −3.61311478277041515588961877950, −2.50724894329374368459203540098, −1.04168621325688380908447018983,
1.04168621325688380908447018983, 2.50724894329374368459203540098, 3.61311478277041515588961877950, 4.25172242842989871526133469504, 5.36249482889116442170023690841, 6.39849465748652448924567504195, 7.47763526176816074486193675448, 7.972516854397409118504829535035, 8.645485390247991723199556988296, 9.532192059495699965382007757015
