L(s) = 1 | + (−0.557 − 0.321i)5-s − 2.15·7-s + 2.27i·11-s + (−0.313 + 0.181i)13-s + (−1.40 − 0.813i)17-s + (2.64 − 3.46i)19-s + (5.64 − 3.25i)23-s + (−2.29 − 3.97i)25-s + (0.968 + 1.67i)29-s + 6.67i·31-s + (1.19 + 0.692i)35-s + 3.30i·37-s + (−3.49 + 6.05i)41-s + (−2.71 + 4.70i)43-s + (7.69 − 4.44i)47-s + ⋯ |
L(s) = 1 | + (−0.249 − 0.143i)5-s − 0.813·7-s + 0.684i·11-s + (−0.0869 + 0.0502i)13-s + (−0.341 − 0.197i)17-s + (0.606 − 0.795i)19-s + (1.17 − 0.679i)23-s + (−0.458 − 0.794i)25-s + (0.179 + 0.311i)29-s + 1.19i·31-s + (0.202 + 0.117i)35-s + 0.544i·37-s + (−0.545 + 0.945i)41-s + (−0.414 + 0.717i)43-s + (1.12 − 0.648i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0622 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0622 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9499210396\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9499210396\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-2.64 + 3.46i)T \) |
good | 5 | \( 1 + (0.557 + 0.321i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 - 2.27iT - 11T^{2} \) |
| 13 | \( 1 + (0.313 - 0.181i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.40 + 0.813i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-5.64 + 3.25i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.968 - 1.67i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.67iT - 31T^{2} \) |
| 37 | \( 1 - 3.30iT - 37T^{2} \) |
| 41 | \( 1 + (3.49 - 6.05i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.71 - 4.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.69 + 4.44i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.96 - 3.40i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.93 - 5.08i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.91 + 6.78i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.4 - 6.01i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.60 - 13.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.63 - 13.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.77 + 1.60i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.6iT - 83T^{2} \) |
| 89 | \( 1 + (-8.16 - 14.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.37 + 4.25i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.952416665443771744244056595225, −8.435454500323395085386392584354, −7.25718083390552252194377639953, −6.91094687818951303846916273181, −6.05838981098087394204404606274, −4.94787302647308945952909261517, −4.44234135737325194617124743082, −3.24248028480935169129801803279, −2.57307869871087670363892383624, −1.10984711306556239010798920903,
0.34304620512433182976511693944, 1.80755041444206623551251189586, 3.15584881133971987501090113909, 3.56250597106241638539310363284, 4.67409595808202992769283626258, 5.76433852595363110700934984180, 6.15642883757777633749393745627, 7.33360678205424658876233000000, 7.62665344020950955763681527356, 8.831945306694003138369664271071