L(s) = 1 | + (0.707 + 0.707i)5-s − 3.83·7-s + (−0.214 + 0.214i)11-s + (−0.177 − 0.177i)13-s + 1.29i·17-s + (−2.10 + 2.10i)19-s − 3.78i·23-s + 1.00i·25-s + (3.15 − 3.15i)29-s − 0.745i·31-s + (−2.71 − 2.71i)35-s + (7.10 − 7.10i)37-s − 10.5·41-s + (2.04 + 2.04i)43-s + 10.9·47-s + ⋯ |
L(s) = 1 | + (0.316 + 0.316i)5-s − 1.45·7-s + (−0.0647 + 0.0647i)11-s + (−0.0493 − 0.0493i)13-s + 0.314i·17-s + (−0.483 + 0.483i)19-s − 0.789i·23-s + 0.200i·25-s + (0.585 − 0.585i)29-s − 0.133i·31-s + (−0.458 − 0.458i)35-s + (1.16 − 1.16i)37-s − 1.64·41-s + (0.311 + 0.311i)43-s + 1.59·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.119870148\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.119870148\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 + (0.214 - 0.214i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.177 + 0.177i)T + 13iT^{2} \) |
| 17 | \( 1 - 1.29iT - 17T^{2} \) |
| 19 | \( 1 + (2.10 - 2.10i)T - 19iT^{2} \) |
| 23 | \( 1 + 3.78iT - 23T^{2} \) |
| 29 | \( 1 + (-3.15 + 3.15i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.745iT - 31T^{2} \) |
| 37 | \( 1 + (-7.10 + 7.10i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + (-2.04 - 2.04i)T + 43iT^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + (0.467 + 0.467i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.61 - 1.61i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.34 - 1.34i)T + 61iT^{2} \) |
| 67 | \( 1 + (-6.92 + 6.92i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.46iT - 71T^{2} \) |
| 73 | \( 1 + 7.24iT - 73T^{2} \) |
| 79 | \( 1 + 12.5iT - 79T^{2} \) |
| 83 | \( 1 + (9.42 + 9.42i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.41T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.786159058241650070673764721094, −7.83006043375579429033293386125, −7.02768621168150600795862791499, −6.20793197830173550073119249189, −5.93895370827611172008535959479, −4.66137918400431276799180274370, −3.76197851984899810724612239597, −2.93944846243232840629127233000, −2.06836767316956857267081079344, −0.41725778174944552479294855855,
0.984074201623134471760161918249, 2.42630699915717059330996298519, 3.20440614386108554242842997952, 4.12410022068169985177353293005, 5.12315012904230687147346142424, 5.88909666056377518656591712152, 6.68371817307060477004396592665, 7.17322887291911604699569633699, 8.335853647571076905668173712639, 8.921877427146971636900487986631