| L(s) = 1 | + 4·2-s − 13·3-s + 16·4-s − 51·5-s − 52·6-s + 105·7-s + 64·8-s − 74·9-s − 204·10-s + 120·11-s − 208·12-s + 420·14-s + 663·15-s + 256·16-s + 1.10e3·17-s − 296·18-s + 1.17e3·19-s − 816·20-s − 1.36e3·21-s + 480·22-s − 1.05e3·23-s − 832·24-s − 524·25-s + 4.12e3·27-s + 1.68e3·28-s − 4.10e3·29-s + 2.65e3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.833·3-s + 1/2·4-s − 0.912·5-s − 0.589·6-s + 0.809·7-s + 0.353·8-s − 0.304·9-s − 0.645·10-s + 0.299·11-s − 0.416·12-s + 0.572·14-s + 0.760·15-s + 1/4·16-s + 0.923·17-s − 0.215·18-s + 0.743·19-s − 0.456·20-s − 0.675·21-s + 0.211·22-s − 0.413·23-s − 0.294·24-s − 0.167·25-s + 1.08·27-s + 0.404·28-s − 0.906·29-s + 0.537·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 13 T + p^{5} T^{2} \) |
| 5 | \( 1 + 51 T + p^{5} T^{2} \) |
| 7 | \( 1 - 15 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 120 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1101 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1170 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1050 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4104 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9624 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8709 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9480 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9995 T + p^{5} T^{2} \) |
| 47 | \( 1 + 2943 T + p^{5} T^{2} \) |
| 53 | \( 1 + 750 T + p^{5} T^{2} \) |
| 59 | \( 1 + 40938 T + p^{5} T^{2} \) |
| 61 | \( 1 + 57920 T + p^{5} T^{2} \) |
| 67 | \( 1 + 22812 T + p^{5} T^{2} \) |
| 71 | \( 1 + 63741 T + p^{5} T^{2} \) |
| 73 | \( 1 - 58866 T + p^{5} T^{2} \) |
| 79 | \( 1 - 63202 T + p^{5} T^{2} \) |
| 83 | \( 1 + 55458 T + p^{5} T^{2} \) |
| 89 | \( 1 + 104778 T + p^{5} T^{2} \) |
| 97 | \( 1 + 160452 T + p^{5} T^{2} \) |
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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
−10.78623719283394712959451018866, −9.383209162677263808848474886327, −7.993905243783836353202329108871, −7.40425234417494959012979037898, −6.03727396277566936603593759221, −5.29792682222680321341512678255, −4.27989320547716546336214365668, −3.21498965374046241410333201043, −1.47652638879185012395105112177, 0,
1.47652638879185012395105112177, 3.21498965374046241410333201043, 4.27989320547716546336214365668, 5.29792682222680321341512678255, 6.03727396277566936603593759221, 7.40425234417494959012979037898, 7.993905243783836353202329108871, 9.383209162677263808848474886327, 10.78623719283394712959451018866
