| L(s) = 1 | + 3.30·2-s + 2.94·4-s + 9.53·5-s − 3.95·7-s − 16.7·8-s + 31.5·10-s + 54.8·11-s − 65.0·13-s − 13.0·14-s − 78.8·16-s − 92.2·17-s + 29.3·19-s + 28.0·20-s + 181.·22-s + 155.·23-s − 34.1·25-s − 215.·26-s − 11.6·28-s + 134.·29-s − 160.·31-s − 127.·32-s − 305.·34-s − 37.7·35-s − 265.·37-s + 97.2·38-s − 159.·40-s + 144.·41-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.367·4-s + 0.852·5-s − 0.213·7-s − 0.739·8-s + 0.997·10-s + 1.50·11-s − 1.38·13-s − 0.249·14-s − 1.23·16-s − 1.31·17-s + 0.354·19-s + 0.313·20-s + 1.75·22-s + 1.40·23-s − 0.273·25-s − 1.62·26-s − 0.0785·28-s + 0.861·29-s − 0.927·31-s − 0.702·32-s − 1.53·34-s − 0.182·35-s − 1.17·37-s + 0.415·38-s − 0.630·40-s + 0.550·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1899 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1899 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 211 | \( 1 + 211T \) |
| good | 2 | \( 1 - 3.30T + 8T^{2} \) |
| 5 | \( 1 - 9.53T + 125T^{2} \) |
| 7 | \( 1 + 3.95T + 343T^{2} \) |
| 11 | \( 1 - 54.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 92.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 160.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 265.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 144.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 332.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 28.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 312.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 377.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 344.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 982.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 996.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 238.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 72.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 390.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 402.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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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
−8.846554027674585889892912798448, −7.25782936244989887591766355225, −6.64813114823875766124138561002, −6.00672295035819271640434216148, −5.03343598608583473184483596172, −4.53570948526566469872668229320, −3.49941358668332366190910191206, −2.62360158085332372273725051456, −1.59917528863286798756577864933, 0,
1.59917528863286798756577864933, 2.62360158085332372273725051456, 3.49941358668332366190910191206, 4.53570948526566469872668229320, 5.03343598608583473184483596172, 6.00672295035819271640434216148, 6.64813114823875766124138561002, 7.25782936244989887591766355225, 8.846554027674585889892912798448
