| L(s) = 1 | + 3-s − 2·5-s + 2·7-s + 9-s − 6·11-s − 13-s − 2·15-s − 2·17-s − 6·19-s + 2·21-s − 25-s + 27-s − 6·29-s + 6·31-s − 6·33-s − 4·35-s + 2·37-s − 39-s − 10·41-s + 8·43-s − 2·45-s + 6·47-s − 3·49-s − 2·51-s + 6·53-s + 12·55-s − 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s − 0.516·15-s − 0.485·17-s − 1.37·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.07·31-s − 1.04·33-s − 0.676·35-s + 0.328·37-s − 0.160·39-s − 1.56·41-s + 1.21·43-s − 0.298·45-s + 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 1.61·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.092559319520639094989226590241, −8.267121982133789942052976540936, −7.84278157109124886213393953192, −7.15914697119095734461341250765, −5.86780953840629681522673799950, −4.78263033612980466766564281150, −4.17228712390106732432301137169, −2.92133996435419900993581940697, −2.01074202882877077393828448729, 0,
2.01074202882877077393828448729, 2.92133996435419900993581940697, 4.17228712390106732432301137169, 4.78263033612980466766564281150, 5.86780953840629681522673799950, 7.15914697119095734461341250765, 7.84278157109124886213393953192, 8.267121982133789942052976540936, 9.092559319520639094989226590241
