| L(s) = 1 | + 2-s + 4-s − 2.82·5-s + 8-s − 2.82·10-s − 11-s + 4.24·13-s + 16-s + 2.82·17-s + 1.41·19-s − 2.82·20-s − 22-s − 6·23-s + 3.00·25-s + 4.24·26-s − 8·29-s − 1.41·31-s + 32-s + 2.82·34-s − 6·37-s + 1.41·38-s − 2.82·40-s − 8.48·41-s + 10·43-s − 44-s − 6·46-s + 7.07·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.26·5-s + 0.353·8-s − 0.894·10-s − 0.301·11-s + 1.17·13-s + 0.250·16-s + 0.685·17-s + 0.324·19-s − 0.632·20-s − 0.213·22-s − 1.25·23-s + 0.600·25-s + 0.832·26-s − 1.48·29-s − 0.254·31-s + 0.176·32-s + 0.485·34-s − 0.986·37-s + 0.229·38-s − 0.447·40-s − 1.32·41-s + 1.52·43-s − 0.150·44-s − 0.884·46-s + 1.03·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + 1.41T + 97T^{2} \) |
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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
−7.41778712323278118655027061120, −6.66121329136082558814821834478, −5.78311394837978738967266114018, −5.37713483173816357730568533469, −4.36089896266003207325585601039, −3.68671553357116794443664839923, −3.48578985938953674834555666432, −2.30936118986711341920204592261, −1.27868173372719731314821292033, 0,
1.27868173372719731314821292033, 2.30936118986711341920204592261, 3.48578985938953674834555666432, 3.68671553357116794443664839923, 4.36089896266003207325585601039, 5.37713483173816357730568533469, 5.78311394837978738967266114018, 6.66121329136082558814821834478, 7.41778712323278118655027061120
