L(s) = 1 | + (−0.5 + 0.866i)5-s + (2 + 3.46i)7-s + (1 + 1.73i)11-s + (−2 + 3.46i)13-s − 17-s − 5·19-s + (2.5 − 4.33i)23-s + (−0.499 − 0.866i)25-s + (4 + 6.92i)29-s + (−3.5 + 6.06i)31-s − 3.99·35-s − 6·37-s + (3 − 5.19i)41-s + (1 + 1.73i)43-s + (4 + 6.92i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.755 + 1.30i)7-s + (0.301 + 0.522i)11-s + (−0.554 + 0.960i)13-s − 0.242·17-s − 1.14·19-s + (0.521 − 0.902i)23-s + (−0.0999 − 0.173i)25-s + (0.742 + 1.28i)29-s + (−0.628 + 1.08i)31-s − 0.676·35-s − 0.986·37-s + (0.468 − 0.811i)41-s + (0.152 + 0.264i)43-s + (0.583 + 1.01i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.182570213\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182570213\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + (-2.5 + 4.33i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4 - 6.92i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + (4.5 + 7.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.5 + 14.7i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (-4 - 6.92i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.874672441891285683138363408087, −8.459835713836355600206200583950, −7.41966874323242114008345338954, −6.73282750453757397433917084721, −6.10205378366196811075483769524, −4.86674180337396369879507346984, −4.68262680522268959672663001683, −3.37923914257925414098751370188, −2.34094684624041533279095256116, −1.72338794199229094653573514545,
0.36002440305948838818549975707, 1.35266748513216560630302293880, 2.61028784156667144848012063859, 3.82663791915193161632532984768, 4.31328079181721841188198261513, 5.18922832595218440742070616850, 6.00645260401279448955405386647, 7.01467121302289410494195635032, 7.64159108066045231990549051785, 8.220243529571390131186645929268