| L(s) = 1 | − 0.618·2-s + 3-s − 1.61·4-s − 0.618·6-s − 2·7-s + 2.23·8-s + 9-s − 3·11-s − 1.61·12-s − 13-s + 1.23·14-s + 1.85·16-s + 0.236·17-s − 0.618·18-s + 6.70·19-s − 2·21-s + 1.85·22-s + 7.61·23-s + 2.23·24-s + 0.618·26-s + 27-s + 3.23·28-s − 1.38·29-s − 4.70·31-s − 5.61·32-s − 3·33-s − 0.145·34-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.577·3-s − 0.809·4-s − 0.252·6-s − 0.755·7-s + 0.790·8-s + 0.333·9-s − 0.904·11-s − 0.467·12-s − 0.277·13-s + 0.330·14-s + 0.463·16-s + 0.0572·17-s − 0.145·18-s + 1.53·19-s − 0.436·21-s + 0.395·22-s + 1.58·23-s + 0.456·24-s + 0.121·26-s + 0.192·27-s + 0.611·28-s − 0.256·29-s − 0.845·31-s − 0.993·32-s − 0.522·33-s − 0.0250·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 9.61T + 43T^{2} \) |
| 47 | \( 1 + 9.23T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 + 1.09T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−8.941544561304888954909567596488, −8.139730978028488726897058985380, −7.45412558900011014759405657885, −6.72668969607071447382370431822, −5.29858645072028312578267372402, −4.92915962404746552943019094001, −3.52621189195649390647901322075, −3.02564765773836666571548989093, −1.49019527194744572704569608914, 0,
1.49019527194744572704569608914, 3.02564765773836666571548989093, 3.52621189195649390647901322075, 4.92915962404746552943019094001, 5.29858645072028312578267372402, 6.72668969607071447382370431822, 7.45412558900011014759405657885, 8.139730978028488726897058985380, 8.941544561304888954909567596488
