L(s) = 1 | − 2.04·2-s + (−0.965 − 1.43i)3-s + 2.18·4-s + (2.23 − 0.144i)5-s + (1.97 + 2.94i)6-s − 0.376·8-s + (−1.13 + 2.77i)9-s + (−4.56 + 0.294i)10-s − 5.15i·11-s + (−2.10 − 3.14i)12-s − 2.98·13-s + (−2.36 − 3.07i)15-s − 3.59·16-s − 1.35i·17-s + (2.32 − 5.67i)18-s + 3.09i·19-s + ⋯ |
L(s) = 1 | − 1.44·2-s + (−0.557 − 0.830i)3-s + 1.09·4-s + (0.997 − 0.0644i)5-s + (0.806 + 1.20i)6-s − 0.133·8-s + (−0.378 + 0.925i)9-s + (−1.44 + 0.0931i)10-s − 1.55i·11-s + (−0.608 − 0.906i)12-s − 0.826·13-s + (−0.609 − 0.792i)15-s − 0.899·16-s − 0.329i·17-s + (0.547 − 1.33i)18-s + 0.709i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.210379 - 0.488833i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.210379 - 0.488833i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 + 1.43i)T \) |
| 5 | \( 1 + (-2.23 + 0.144i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 11 | \( 1 + 5.15iT - 11T^{2} \) |
| 13 | \( 1 + 2.98T + 13T^{2} \) |
| 17 | \( 1 + 1.35iT - 17T^{2} \) |
| 19 | \( 1 - 3.09iT - 19T^{2} \) |
| 23 | \( 1 - 7.22T + 23T^{2} \) |
| 29 | \( 1 + 2.69iT - 29T^{2} \) |
| 31 | \( 1 + 4.35iT - 31T^{2} \) |
| 37 | \( 1 + 7.59iT - 37T^{2} \) |
| 41 | \( 1 + 7.68T + 41T^{2} \) |
| 43 | \( 1 - 7.79iT - 43T^{2} \) |
| 47 | \( 1 + 6.07iT - 47T^{2} \) |
| 53 | \( 1 + 6.29T + 53T^{2} \) |
| 59 | \( 1 + 4.25T + 59T^{2} \) |
| 61 | \( 1 + 5.21iT - 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 11.2iT - 71T^{2} \) |
| 73 | \( 1 + 4.91T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 9.32T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−9.969852105837904851827844052644, −9.185628425471609132175321741884, −8.401039209673167501549807364534, −7.60327113584094083269967653670, −6.71288066452824112189454517337, −5.90458379487019826448830417632, −4.98454837974845147154393353512, −2.82285955757733431935886322308, −1.66661280945474583144617446138, −0.50253549698247450359974822406,
1.44295509308917114973767125195, 2.77148103635708614554245709097, 4.64436128312155318495998834866, 5.18857925596531382450152656903, 6.73022597829161649528527164803, 7.07189858169309399354167815265, 8.504899889329225749322856556751, 9.343769171768939651651562721391, 9.721186679194833586539393876718, 10.42542156668367978181718022743