| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s − 2·13-s + 14-s + 16-s − 2·17-s − 20-s − 22-s − 6·23-s + 25-s − 2·26-s + 28-s − 8·29-s − 8·31-s + 32-s − 2·34-s − 35-s + 2·37-s − 40-s − 2·41-s − 44-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s − 0.213·22-s − 1.25·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.48·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s + 0.328·37-s − 0.158·40-s − 0.312·41-s − 0.150·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9708588512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9708588512\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.31340756542175, −12.66702438611928, −12.43488630947225, −11.83426298786712, −11.36440889985331, −11.02545130289963, −10.58902730989609, −9.913416045069219, −9.464187459378830, −8.960003652994285, −8.245323791609431, −7.761872095782316, −7.550060045340160, −6.857532647358463, −6.374795182957994, −5.771530971051678, −5.261373754515611, −4.810294885130011, −4.265589533953920, −3.674643827567892, −3.327310829381605, −2.435433647869767, −2.004057212156091, −1.425038012951777, −0.2350014062906119,
0.2350014062906119, 1.425038012951777, 2.004057212156091, 2.435433647869767, 3.327310829381605, 3.674643827567892, 4.265589533953920, 4.810294885130011, 5.261373754515611, 5.771530971051678, 6.374795182957994, 6.857532647358463, 7.550060045340160, 7.761872095782316, 8.245323791609431, 8.960003652994285, 9.464187459378830, 9.913416045069219, 10.58902730989609, 11.02545130289963, 11.36440889985331, 11.83426298786712, 12.43488630947225, 12.66702438611928, 13.31340756542175
