| L(s) = 1 | + 2-s + 4-s + 5-s + 4.33·7-s + 8-s + 10-s + 4.74·11-s + 0.409·13-s + 4.33·14-s + 16-s − 5.77·17-s − 5.77·19-s + 20-s + 4.74·22-s + 9.36·23-s + 25-s + 0.409·26-s + 4.33·28-s + 1.30·29-s + 2.40·31-s + 32-s − 5.77·34-s + 4.33·35-s + 3.02·37-s − 5.77·38-s + 40-s − 1.71·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.63·7-s + 0.353·8-s + 0.316·10-s + 1.43·11-s + 0.113·13-s + 1.15·14-s + 0.250·16-s − 1.40·17-s − 1.32·19-s + 0.223·20-s + 1.01·22-s + 1.95·23-s + 0.200·25-s + 0.0802·26-s + 0.819·28-s + 0.242·29-s + 0.432·31-s + 0.176·32-s − 0.990·34-s + 0.732·35-s + 0.497·37-s − 0.936·38-s + 0.158·40-s − 0.268·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.856914599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.856914599\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 4.33T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 0.409T + 13T^{2} \) |
| 17 | \( 1 + 5.77T + 17T^{2} \) |
| 19 | \( 1 + 5.77T + 19T^{2} \) |
| 23 | \( 1 - 9.36T + 23T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 - 2.40T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 + 1.71T + 41T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 - 1.51T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 71 | \( 1 + 8.11T + 71T^{2} \) |
| 73 | \( 1 + 8.05T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 + 8.57T + 83T^{2} \) |
| 89 | \( 1 - 9.08T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−8.198669804035354407981298936259, −6.98924109777514627592385074383, −6.75788608481903978949861074201, −5.90055250539786635375712053750, −4.99715441297207004980174738992, −4.52128669086471931970452027593, −3.92593429429863213739557066390, −2.69559758127604620805874421086, −1.88089965247911845607076969805, −1.16785206999067924713004166765,
1.16785206999067924713004166765, 1.88089965247911845607076969805, 2.69559758127604620805874421086, 3.92593429429863213739557066390, 4.52128669086471931970452027593, 4.99715441297207004980174738992, 5.90055250539786635375712053750, 6.75788608481903978949861074201, 6.98924109777514627592385074383, 8.198669804035354407981298936259
