| L(s) = 1 | − 2-s + 4-s − 3·5-s − 7-s − 8-s + 3·10-s − 2·13-s + 14-s + 16-s + 17-s − 7·19-s − 3·20-s − 23-s + 4·25-s + 2·26-s − 28-s − 8·29-s − 9·31-s − 32-s − 34-s + 3·35-s − 5·37-s + 7·38-s + 3·40-s − 2·41-s + 8·43-s + 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.60·19-s − 0.670·20-s − 0.208·23-s + 4/5·25-s + 0.392·26-s − 0.188·28-s − 1.48·29-s − 1.61·31-s − 0.176·32-s − 0.171·34-s + 0.507·35-s − 0.821·37-s + 1.13·38-s + 0.474·40-s − 0.312·41-s + 1.21·43-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−12.89977147266936, −12.49206318678246, −12.16315693359661, −11.51048239102698, −11.19180272790533, −10.81457588077470, −10.28703237405678, −9.838017932132892, −9.183169628513278, −8.901011172672967, −8.375303125500257, −7.907541772239300, −7.401667606140583, −7.137509659181080, −6.672310541170971, −5.794232101750776, −5.691315899343856, −4.744035538978342, −4.223649688546097, −3.821125345836239, −3.272903338699546, −2.696059073238789, −1.944697508932161, −1.504837624257900, −0.3691806006246032, 0,
0.3691806006246032, 1.504837624257900, 1.944697508932161, 2.696059073238789, 3.272903338699546, 3.821125345836239, 4.223649688546097, 4.744035538978342, 5.691315899343856, 5.794232101750776, 6.672310541170971, 7.137509659181080, 7.401667606140583, 7.907541772239300, 8.375303125500257, 8.901011172672967, 9.183169628513278, 9.838017932132892, 10.28703237405678, 10.81457588077470, 11.19180272790533, 11.51048239102698, 12.16315693359661, 12.49206318678246, 12.89977147266936
