| L(s) = 1 | − 2-s − 4-s + 5-s + 2·7-s + 3·8-s − 10-s + 2·11-s − 2·14-s − 16-s + 7·17-s − 6·19-s − 20-s − 2·22-s + 6·23-s − 4·25-s − 2·28-s + 29-s + 4·31-s − 5·32-s − 7·34-s + 2·35-s + 37-s + 6·38-s + 3·40-s − 9·41-s + 6·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s + 1.06·8-s − 0.316·10-s + 0.603·11-s − 0.534·14-s − 1/4·16-s + 1.69·17-s − 1.37·19-s − 0.223·20-s − 0.426·22-s + 1.25·23-s − 4/5·25-s − 0.377·28-s + 0.185·29-s + 0.718·31-s − 0.883·32-s − 1.20·34-s + 0.338·35-s + 0.164·37-s + 0.973·38-s + 0.474·40-s − 1.40·41-s + 0.914·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287980953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287980953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.462311395423910043887289212496, −8.637948500833363093095859501567, −8.104199630885793933442425068756, −7.28313700976548615717570646317, −6.24903912820188716994661962896, −5.25576523488268654791848893305, −4.52587188409845770822800829766, −3.48093369409586291685165707364, −1.94863949034463228039186443244, −0.969389822342126780434621231259,
0.969389822342126780434621231259, 1.94863949034463228039186443244, 3.48093369409586291685165707364, 4.52587188409845770822800829766, 5.25576523488268654791848893305, 6.24903912820188716994661962896, 7.28313700976548615717570646317, 8.104199630885793933442425068756, 8.637948500833363093095859501567, 9.462311395423910043887289212496
