| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s + 7-s − 3·8-s + 9-s − 10-s + 6·11-s − 12-s + 14-s − 15-s − 16-s − 17-s + 18-s − 4·19-s + 20-s + 21-s + 6·22-s − 6·23-s − 3·24-s + 25-s + 27-s − 28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s + 1.27·22-s − 1.25·23-s − 0.612·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.73599714738834, −12.58805461163304, −12.04568238183241, −11.77472237129138, −11.18101744987881, −10.72649905531963, −10.07905718697218, −9.579664096506045, −9.094546667955510, −8.849397388522927, −8.362964070401076, −7.843711584196564, −7.425876393526772, −6.608267256741226, −6.332094863577021, −5.948192511195821, −5.180030503277658, −4.500309454070926, −4.318792647028763, −3.895655813589098, −3.401310230986106, −2.847882273141425, −2.076244171323333, −1.562785782489115, −0.8155537925615644, 0,
0.8155537925615644, 1.562785782489115, 2.076244171323333, 2.847882273141425, 3.401310230986106, 3.895655813589098, 4.318792647028763, 4.500309454070926, 5.180030503277658, 5.948192511195821, 6.332094863577021, 6.608267256741226, 7.425876393526772, 7.843711584196564, 8.362964070401076, 8.849397388522927, 9.094546667955510, 9.579664096506045, 10.07905718697218, 10.72649905531963, 11.18101744987881, 11.77472237129138, 12.04568238183241, 12.58805461163304, 12.73599714738834
