| L(s) = 1 | − 7.87·3-s − 13.0·5-s − 35.1·7-s + 35.0·9-s − 66.4·11-s + 40.5·13-s + 103.·15-s + 39.8·17-s − 93.1·19-s + 276.·21-s − 23·23-s + 46.1·25-s − 63.0·27-s − 234.·29-s + 226.·31-s + 523.·33-s + 459.·35-s + 275.·37-s − 318.·39-s + 59.0·41-s − 184.·43-s − 457.·45-s + 506.·47-s + 889.·49-s − 313.·51-s + 8.25·53-s + 869.·55-s + ⋯ |
| L(s) = 1 | − 1.51·3-s − 1.17·5-s − 1.89·7-s + 1.29·9-s − 1.82·11-s + 0.864·13-s + 1.77·15-s + 0.568·17-s − 1.12·19-s + 2.87·21-s − 0.208·23-s + 0.369·25-s − 0.449·27-s − 1.50·29-s + 1.31·31-s + 2.76·33-s + 2.21·35-s + 1.22·37-s − 1.30·39-s + 0.224·41-s − 0.652·43-s − 1.51·45-s + 1.57·47-s + 2.59·49-s − 0.861·51-s + 0.0213·53-s + 2.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 7.87T + 27T^{2} \) |
| 5 | \( 1 + 13.0T + 125T^{2} \) |
| 7 | \( 1 + 35.1T + 343T^{2} \) |
| 11 | \( 1 + 66.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 226.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 275.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 59.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 184.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 506.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 8.25T + 1.48e5T^{2} \) |
| 59 | \( 1 - 483.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 411.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 324.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 752.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 136.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 944.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 271.T + 9.12e5T^{2} \) |
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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
−8.669910736923300202928062480155, −7.73399009004740602842935110567, −7.02265604759270307343278668882, −6.03726470406550353149570859634, −5.75083290751400005618936916304, −4.51309614775891442840290743931, −3.68627005356804579590516419731, −2.69766523348999238617496922337, −0.62261733248286535705815709966, 0,
0.62261733248286535705815709966, 2.69766523348999238617496922337, 3.68627005356804579590516419731, 4.51309614775891442840290743931, 5.75083290751400005618936916304, 6.03726470406550353149570859634, 7.02265604759270307343278668882, 7.73399009004740602842935110567, 8.669910736923300202928062480155
