| L(s) = 1 | + 2·2-s − 2.88·3-s + 4·4-s − 5.77·6-s − 8.21·7-s + 8·8-s − 18.6·9-s − 71.5·11-s − 11.5·12-s − 66.1·13-s − 16.4·14-s + 16·16-s − 27.7·17-s − 37.3·18-s + 13.7·19-s + 23.7·21-s − 143.·22-s + 23·23-s − 23.1·24-s − 132.·26-s + 131.·27-s − 32.8·28-s + 274.·29-s + 265.·31-s + 32·32-s + 206.·33-s − 55.5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.555·3-s + 0.5·4-s − 0.393·6-s − 0.443·7-s + 0.353·8-s − 0.690·9-s − 1.96·11-s − 0.277·12-s − 1.41·13-s − 0.313·14-s + 0.250·16-s − 0.396·17-s − 0.488·18-s + 0.166·19-s + 0.246·21-s − 1.38·22-s + 0.208·23-s − 0.196·24-s − 0.997·26-s + 0.940·27-s − 0.221·28-s + 1.75·29-s + 1.54·31-s + 0.176·32-s + 1.09·33-s − 0.280·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.384915064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.384915064\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 2.88T + 27T^{2} \) |
| 7 | \( 1 + 8.21T + 343T^{2} \) |
| 11 | \( 1 + 71.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 66.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 274.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 265.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 204.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 69.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 187.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 135.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 502.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 161.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 103.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 985.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 817.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 400.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 485.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−9.692146552976899456816354988580, −8.397872902682516461748086828448, −7.67906167473951001513603375903, −6.73583260371538783511906904832, −5.90391198397662059541023950416, −5.07769183511187301159499352370, −4.56431139843586896767003405770, −2.80281516551701884565744269564, −2.63596842338367123260334931880, −0.52507325521308915428296060630,
0.52507325521308915428296060630, 2.63596842338367123260334931880, 2.80281516551701884565744269564, 4.56431139843586896767003405770, 5.07769183511187301159499352370, 5.90391198397662059541023950416, 6.73583260371538783511906904832, 7.67906167473951001513603375903, 8.397872902682516461748086828448, 9.692146552976899456816354988580
