| L(s) = 1 | + 2-s + 0.529·3-s + 4-s − 5-s + 0.529·6-s − 4.65·7-s + 8-s − 2.71·9-s − 10-s − 5.29·11-s + 0.529·12-s − 4.82·13-s − 4.65·14-s − 0.529·15-s + 16-s + 5.84·17-s − 2.71·18-s + 7.01·19-s − 20-s − 2.46·21-s − 5.29·22-s + 3.32·23-s + 0.529·24-s + 25-s − 4.82·26-s − 3.03·27-s − 4.65·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.305·3-s + 0.5·4-s − 0.447·5-s + 0.216·6-s − 1.75·7-s + 0.353·8-s − 0.906·9-s − 0.316·10-s − 1.59·11-s + 0.152·12-s − 1.33·13-s − 1.24·14-s − 0.136·15-s + 0.250·16-s + 1.41·17-s − 0.640·18-s + 1.60·19-s − 0.223·20-s − 0.538·21-s − 1.12·22-s + 0.693·23-s + 0.108·24-s + 0.200·25-s − 0.946·26-s − 0.583·27-s − 0.879·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.692455047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.692455047\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 0.529T + 3T^{2} \) |
| 7 | \( 1 + 4.65T + 7T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 5.84T + 17T^{2} \) |
| 19 | \( 1 - 7.01T + 19T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 - 7.14T + 29T^{2} \) |
| 31 | \( 1 - 8.39T + 31T^{2} \) |
| 37 | \( 1 - 8.13T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 6.06T + 43T^{2} \) |
| 47 | \( 1 - 0.578T + 47T^{2} \) |
| 53 | \( 1 - 0.566T + 53T^{2} \) |
| 59 | \( 1 + 6.22T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 9.02T + 73T^{2} \) |
| 79 | \( 1 + 7.60T + 79T^{2} \) |
| 83 | \( 1 - 5.57T + 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 - 4.67T + 97T^{2} \) |
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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
−8.123960731493211003693673096800, −7.75205515621435650651731622019, −6.93630964935513735093878751820, −6.19741109440303671165066793823, −5.26783720885087538403982817143, −4.93804816743450102779832757903, −3.49226904142777383895273721270, −2.86894062307337482593568772299, −2.76693115207564487999698111665, −0.61851491911195767634210066350,
0.61851491911195767634210066350, 2.76693115207564487999698111665, 2.86894062307337482593568772299, 3.49226904142777383895273721270, 4.93804816743450102779832757903, 5.26783720885087538403982817143, 6.19741109440303671165066793823, 6.93630964935513735093878751820, 7.75205515621435650651731622019, 8.123960731493211003693673096800
