L(s) = 1 | + (−1 − 1.73i)3-s + (0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s − 5·11-s + (1 − 1.73i)13-s + (0.999 − 1.73i)15-s + (1 + 1.73i)17-s + (0.5 − 4.33i)19-s + (−2 + 3.46i)23-s + (−0.499 + 0.866i)25-s − 4.00·27-s + (−4.5 + 7.79i)29-s + 5·31-s + (5 + 8.66i)33-s − 4·37-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.999i)3-s + (0.223 + 0.387i)5-s + (−0.166 + 0.288i)9-s − 1.50·11-s + (0.277 − 0.480i)13-s + (0.258 − 0.447i)15-s + (0.242 + 0.420i)17-s + (0.114 − 0.993i)19-s + (−0.417 + 0.722i)23-s + (−0.0999 + 0.173i)25-s − 0.769·27-s + (−0.835 + 1.44i)29-s + 0.898·31-s + (0.870 + 1.50i)33-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 4.33i)T \) |
good | 3 | \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (5 + 8.66i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.5 - 6.06i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.828149591310003789094087159934, −7.951689218856180085238444540306, −7.24284723377258366259614155675, −6.65654893556439285604750135522, −5.59530030603036957788688274146, −5.21585047034488577129244548789, −3.63381767495249689652536005591, −2.60364872770148818244275312678, −1.46982861166947609090705224929, 0,
1.89812002987390005475349983760, 3.19580051175721834996511101896, 4.34472758221496913229938292379, 4.95929844191466914755740187129, 5.68932970102090454328559427328, 6.49938408932903444026920345928, 7.88394277762753307371168365857, 8.227569833177510458299000045259, 9.556506727158343040850581969761