| L(s) = 1 | − 0.381·2-s − 3-s − 1.85·4-s + 0.381·6-s + 7-s + 1.47·8-s + 9-s + 3.47·11-s + 1.85·12-s − 5.23·13-s − 0.381·14-s + 3.14·16-s − 5.70·17-s − 0.381·18-s + 1.23·19-s − 21-s − 1.32·22-s − 5·23-s − 1.47·24-s + 2·26-s − 27-s − 1.85·28-s − 8.70·29-s − 4.47·31-s − 4.14·32-s − 3.47·33-s + 2.18·34-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.577·3-s − 0.927·4-s + 0.155·6-s + 0.377·7-s + 0.520·8-s + 0.333·9-s + 1.04·11-s + 0.535·12-s − 1.45·13-s − 0.102·14-s + 0.786·16-s − 1.38·17-s − 0.0900·18-s + 0.283·19-s − 0.218·21-s − 0.282·22-s − 1.04·23-s − 0.300·24-s + 0.392·26-s − 0.192·27-s − 0.350·28-s − 1.61·29-s − 0.803·31-s − 0.732·32-s − 0.604·33-s + 0.373·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 3.76T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 5.23T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 - 6.23T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 - 7.70T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−10.27572180830854116293524091831, −9.514307379143796514462021843763, −8.830529235081570255549226737486, −7.71136916742223777566044488345, −6.84946010032524886166239090026, −5.60355445525633470543539598435, −4.69454518589841104463684318917, −3.88046526236381037132029204139, −1.85382834451832876086335010287, 0,
1.85382834451832876086335010287, 3.88046526236381037132029204139, 4.69454518589841104463684318917, 5.60355445525633470543539598435, 6.84946010032524886166239090026, 7.71136916742223777566044488345, 8.830529235081570255549226737486, 9.514307379143796514462021843763, 10.27572180830854116293524091831
