| L(s) = 1 | − 2-s + 4-s − 1.52·5-s + 0.908·7-s − 8-s + 1.52·10-s − 5.03·11-s − 5.51·13-s − 0.908·14-s + 16-s − 5.38·17-s + 4.02·19-s − 1.52·20-s + 5.03·22-s − 0.290·23-s − 2.67·25-s + 5.51·26-s + 0.908·28-s − 8.72·29-s + 10.2·31-s − 32-s + 5.38·34-s − 1.38·35-s − 6.80·37-s − 4.02·38-s + 1.52·40-s + 6.15·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.681·5-s + 0.343·7-s − 0.353·8-s + 0.482·10-s − 1.51·11-s − 1.52·13-s − 0.242·14-s + 0.250·16-s − 1.30·17-s + 0.923·19-s − 0.340·20-s + 1.07·22-s − 0.0605·23-s − 0.534·25-s + 1.08·26-s + 0.171·28-s − 1.61·29-s + 1.83·31-s − 0.176·32-s + 0.923·34-s − 0.234·35-s − 1.11·37-s − 0.652·38-s + 0.241·40-s + 0.960·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5225677036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5225677036\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 1.52T + 5T^{2} \) |
| 7 | \( 1 - 0.908T + 7T^{2} \) |
| 11 | \( 1 + 5.03T + 11T^{2} \) |
| 13 | \( 1 + 5.51T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 - 4.02T + 19T^{2} \) |
| 23 | \( 1 + 0.290T + 23T^{2} \) |
| 29 | \( 1 + 8.72T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 6.80T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 + 7.03T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 4.35T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 1.04T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 2.46T + 97T^{2} \) |
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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
−8.275522449739939059436548781099, −7.70702836332964771737916788041, −7.35298880434719007098964969672, −6.45788726287495456387668647957, −5.29270749151476989601460812420, −4.86887993543384823062012475949, −3.77966885821998555392943002772, −2.66379588211572041602563066577, −2.07883231118365935514476766449, −0.43052389911182868387624310114,
0.43052389911182868387624310114, 2.07883231118365935514476766449, 2.66379588211572041602563066577, 3.77966885821998555392943002772, 4.86887993543384823062012475949, 5.29270749151476989601460812420, 6.45788726287495456387668647957, 7.35298880434719007098964969672, 7.70702836332964771737916788041, 8.275522449739939059436548781099
