| L(s) = 1 | + 2-s + 4-s + 2·5-s − 2·7-s + 8-s + 2·10-s + 11-s + 13-s − 2·14-s + 16-s + 17-s − 4·19-s + 2·20-s + 22-s + 4·23-s − 25-s + 26-s − 2·28-s + 4·29-s − 6·31-s + 32-s + 34-s − 4·35-s − 2·37-s − 4·38-s + 2·40-s − 12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s + 0.632·10-s + 0.301·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.447·20-s + 0.213·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.377·28-s + 0.742·29-s − 1.07·31-s + 0.176·32-s + 0.171·34-s − 0.676·35-s − 0.328·37-s − 0.648·38-s + 0.316·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.139539416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.139539416\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.55810377067033, −14.08291818740882, −13.62068942643495, −13.09046848418955, −12.87850387516105, −12.03097539479195, −11.88625817978291, −10.88003777101016, −10.58703087572959, −10.07028133697878, −9.419001248211009, −8.971963895927298, −8.400511057768874, −7.607693326994042, −6.837514999999137, −6.633587494135278, −5.915300123587524, −5.548693448737854, −4.859677474952782, −4.177997972320920, −3.483139515372617, −3.014471446435799, −2.153638806242413, −1.668867880791157, −0.6294567253515093,
0.6294567253515093, 1.668867880791157, 2.153638806242413, 3.014471446435799, 3.483139515372617, 4.177997972320920, 4.859677474952782, 5.548693448737854, 5.915300123587524, 6.633587494135278, 6.837514999999137, 7.607693326994042, 8.400511057768874, 8.971963895927298, 9.419001248211009, 10.07028133697878, 10.58703087572959, 10.88003777101016, 11.88625817978291, 12.03097539479195, 12.87850387516105, 13.09046848418955, 13.62068942643495, 14.08291818740882, 14.55810377067033
