| L(s) = 1 | + 2-s + 2.26·3-s + 4-s − 5-s + 2.26·6-s + 3.26·7-s + 8-s + 2.11·9-s − 10-s + 3·11-s + 2.26·12-s − 3.37·13-s + 3.26·14-s − 2.26·15-s + 16-s − 4.49·17-s + 2.11·18-s − 4.52·19-s − 20-s + 7.37·21-s + 3·22-s − 2.52·23-s + 2.26·24-s + 25-s − 3.37·26-s − 2·27-s + 3.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.30·3-s + 0.5·4-s − 0.447·5-s + 0.923·6-s + 1.23·7-s + 0.353·8-s + 0.705·9-s − 0.316·10-s + 0.904·11-s + 0.652·12-s − 0.936·13-s + 0.871·14-s − 0.583·15-s + 0.250·16-s − 1.08·17-s + 0.498·18-s − 1.03·19-s − 0.223·20-s + 1.60·21-s + 0.639·22-s − 0.526·23-s + 0.461·24-s + 0.200·25-s − 0.662·26-s − 0.384·27-s + 0.616·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.440468450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.440468450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 3 | \( 1 - 2.26T + 3T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 + 4.52T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 - 8.90T + 37T^{2} \) |
| 41 | \( 1 - 8.49T + 41T^{2} \) |
| 43 | \( 1 - 1.96T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 3.37T + 53T^{2} \) |
| 59 | \( 1 + 8.49T + 59T^{2} \) |
| 61 | \( 1 + 8.90T + 61T^{2} \) |
| 71 | \( 1 + 0.377T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 6.75T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 9.23T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−10.73509060375669524245629604766, −9.386541499575175627569271567172, −8.697475674874157993778015064095, −7.86651839168610754197063379436, −7.21217982385239196896931177251, −5.99518811950221603646662098174, −4.48366851454844271648500320887, −4.18003864931965464848247036688, −2.75004640854639959274479088489, −1.86572878344884382657876861769,
1.86572878344884382657876861769, 2.75004640854639959274479088489, 4.18003864931965464848247036688, 4.48366851454844271648500320887, 5.99518811950221603646662098174, 7.21217982385239196896931177251, 7.86651839168610754197063379436, 8.697475674874157993778015064095, 9.386541499575175627569271567172, 10.73509060375669524245629604766
