L(s) = 1 | + (0.900 + 0.433i)4-s + (0.222 − 0.974i)7-s + (−1.52 − 0.347i)13-s + (0.623 + 0.781i)16-s + 1.94i·19-s + (−0.222 − 0.974i)25-s + (0.623 − 0.781i)28-s − 0.867i·31-s + (−1.62 + 0.781i)37-s + (0.277 + 0.347i)43-s + (−0.900 − 0.433i)49-s + (−1.22 − 0.974i)52-s + (−0.376 − 0.781i)61-s + (0.222 + 0.974i)64-s + 1.24·67-s + ⋯ |
L(s) = 1 | + (0.900 + 0.433i)4-s + (0.222 − 0.974i)7-s + (−1.52 − 0.347i)13-s + (0.623 + 0.781i)16-s + 1.94i·19-s + (−0.222 − 0.974i)25-s + (0.623 − 0.781i)28-s − 0.867i·31-s + (−1.62 + 0.781i)37-s + (0.277 + 0.347i)43-s + (−0.900 − 0.433i)49-s + (−1.22 − 0.974i)52-s + (−0.376 − 0.781i)61-s + (0.222 + 0.974i)64-s + 1.24·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9913819414\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9913819414\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.222 + 0.974i)T \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 5 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 13 | \( 1 + (1.52 + 0.347i)T + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 - 1.94iT - T^{2} \) |
| 23 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 29 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + 0.867iT - T^{2} \) |
| 37 | \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 61 | \( 1 + (0.376 + 0.781i)T + (-0.623 + 0.781i)T^{2} \) |
| 67 | \( 1 - 1.24T + T^{2} \) |
| 71 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (0.846 - 0.193i)T + (0.900 - 0.433i)T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 + 1.94iT - T^{2} \) |
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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
−11.41201889157146854518793628301, −10.28295572890717336624242386507, −9.987492537363328129202075676935, −8.273247674352147237500971157719, −7.65568389338397475151220458573, −6.88385193308016440535286434908, −5.78104499938859262269552110800, −4.41969686715835042120173225413, −3.29510686126521703266866764348, −1.91746541672123797073516136363,
2.01762822689431474824084583115, 2.93796922257296819669021008800, 4.86668972540395718839892426619, 5.55360292523801343958950518386, 6.84712120383781209023516684014, 7.37199799832499826984190017589, 8.793465225151556040295466600930, 9.504882862774435267936810518790, 10.54048706682459411598847510245, 11.42920279608250369262790393420