| L(s) = 1 | + 3.82·2-s + 3·3-s + 6.65·4-s + 11.4·6-s + 16.8·7-s − 5.14·8-s + 9·9-s − 55.7·11-s + 19.9·12-s + 46.9·13-s + 64.4·14-s − 72.9·16-s − 62.1·17-s + 34.4·18-s − 141.·19-s + 50.4·21-s − 213.·22-s − 23·23-s − 15.4·24-s + 179.·26-s + 27·27-s + 112.·28-s − 288.·29-s + 68.7·31-s − 238.·32-s − 167.·33-s − 237.·34-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.577·3-s + 0.832·4-s + 0.781·6-s + 0.908·7-s − 0.227·8-s + 0.333·9-s − 1.52·11-s + 0.480·12-s + 1.00·13-s + 1.22·14-s − 1.13·16-s − 0.886·17-s + 0.451·18-s − 1.71·19-s + 0.524·21-s − 2.07·22-s − 0.208·23-s − 0.131·24-s + 1.35·26-s + 0.192·27-s + 0.756·28-s − 1.84·29-s + 0.398·31-s − 1.31·32-s − 0.883·33-s − 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 3.82T + 8T^{2} \) |
| 7 | \( 1 - 16.8T + 343T^{2} \) |
| 11 | \( 1 + 55.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 288.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 68.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 71.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 159.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 12.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 426.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 243.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 81.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 696.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 568.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 719.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 337.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−8.499371982951712201977177384550, −7.80612813662392544262441093964, −6.79599050415356389447333356503, −5.88603932685045812321793255650, −5.15096171190085313178987284286, −4.34108341644977490859426905557, −3.70327090202665127754689102481, −2.54426258077692497859776841854, −1.90582980938998709478424543111, 0,
1.90582980938998709478424543111, 2.54426258077692497859776841854, 3.70327090202665127754689102481, 4.34108341644977490859426905557, 5.15096171190085313178987284286, 5.88603932685045812321793255650, 6.79599050415356389447333356503, 7.80612813662392544262441093964, 8.499371982951712201977177384550
