| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s + 11-s − 12-s + 13-s + 2·14-s + 16-s + 17-s + 18-s − 4·19-s − 2·21-s + 22-s + 6·23-s − 24-s + 26-s − 27-s + 2·28-s + 10·31-s + 32-s − 33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.377·28-s + 1.79·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.61554563574509, −12.35576540728275, −11.80318028914216, −11.52826030176318, −10.87211756252221, −10.66259067154515, −10.36220146488002, −9.527868993960733, −9.158852849190628, −8.528924959556472, −8.175117740737940, −7.572773707222101, −7.121124365133986, −6.569418577232248, −6.287496779888464, −5.639825096054508, −5.279457482930314, −4.626327315225669, −4.460207339281565, −3.842641114036795, −3.210975907312969, −2.658305181052765, −2.047058744495871, −1.333335166182877, −0.9950416183914681, 0,
0.9950416183914681, 1.333335166182877, 2.047058744495871, 2.658305181052765, 3.210975907312969, 3.842641114036795, 4.460207339281565, 4.626327315225669, 5.279457482930314, 5.639825096054508, 6.287496779888464, 6.569418577232248, 7.121124365133986, 7.572773707222101, 8.175117740737940, 8.528924959556472, 9.158852849190628, 9.527868993960733, 10.36220146488002, 10.66259067154515, 10.87211756252221, 11.52826030176318, 11.80318028914216, 12.35576540728275, 12.61554563574509
