| L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s − 11-s + 16-s − 17-s − 3·18-s − 5·19-s − 22-s + 2·23-s − 5·25-s − 5·29-s + 8·31-s + 32-s − 34-s − 3·36-s + 8·37-s − 5·38-s − 6·43-s − 44-s + 2·46-s − 11·47-s − 5·50-s + 5·53-s − 5·58-s + 3·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 0.301·11-s + 1/4·16-s − 0.242·17-s − 0.707·18-s − 1.14·19-s − 0.213·22-s + 0.417·23-s − 25-s − 0.928·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s − 1/2·36-s + 1.31·37-s − 0.811·38-s − 0.914·43-s − 0.150·44-s + 0.294·46-s − 1.60·47-s − 0.707·50-s + 0.686·53-s − 0.656·58-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16562 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16562 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.407263264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.407263264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−15.71103116629378, −15.32903722300670, −14.66629691443495, −14.42473015421116, −13.59525673135902, −13.16772240233974, −12.80851420860179, −11.86935806681203, −11.56601355207552, −11.06275527630722, −10.38602425944582, −9.776929449525744, −9.082838405179284, −8.248086685902620, −8.050158757246071, −7.133882180978751, −6.358989871278175, −6.062631618837873, −5.246757593518926, −4.706826573452394, −3.938788910943556, −3.254236058515902, −2.485785072042856, −1.910317194191268, −0.5623580173961588,
0.5623580173961588, 1.910317194191268, 2.485785072042856, 3.254236058515902, 3.938788910943556, 4.706826573452394, 5.246757593518926, 6.062631618837873, 6.358989871278175, 7.133882180978751, 8.050158757246071, 8.248086685902620, 9.082838405179284, 9.776929449525744, 10.38602425944582, 11.06275527630722, 11.56601355207552, 11.86935806681203, 12.80851420860179, 13.16772240233974, 13.59525673135902, 14.42473015421116, 14.66629691443495, 15.32903722300670, 15.71103116629378
