| L(s) = 1 | + 4i·2-s − 17.8i·3-s − 16·4-s + 71.2·6-s − 86.7i·7-s − 64i·8-s − 74.2·9-s − 203.·11-s + 284. i·12-s − 169i·13-s + 347.·14-s + 256·16-s + 406. i·17-s − 296. i·18-s + 2.32e3·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.14i·3-s − 0.5·4-s + 0.807·6-s − 0.669i·7-s − 0.353i·8-s − 0.305·9-s − 0.506·11-s + 0.571i·12-s − 0.277i·13-s + 0.473·14-s + 0.250·16-s + 0.341i·17-s − 0.216i·18-s + 1.47·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.058365645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.058365645\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 3 | \( 1 + 17.8iT - 243T^{2} \) |
| 7 | \( 1 + 86.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 203.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 406. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.32e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.28e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.12e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.50e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.79e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.66e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.72e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.76e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.90e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 8.32e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.94e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 9.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.41e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 2.66e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.49e4iT - 8.58e9T^{2} \) |
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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
−9.572500880889281055573222555283, −8.387278002960991688276223787151, −7.51649438819996965801222050315, −7.28393007395354808916350403611, −6.17959569343121226102340625097, −5.35558898321617319849330852748, −4.14769519482778580216509904244, −2.88868773040646325603838085657, −1.40542989983443627680945384795, −0.58610135300322567807997016850,
0.913920297468290821391087662090, 2.45903098333016851064125675268, 3.24908430692276574402439896256, 4.44575963804332836497249691995, 5.01655846567430697260349276203, 6.06913689864061043906874800797, 7.46953300437329196435540208044, 8.579212389603211898334613537366, 9.305572352320784961918797905132, 9.990687047186043743232888909532
