L(s) = 1 | + (−0.909 + 1.08i)3-s + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)7-s + (−0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.483 − 1.32i)15-s + (0.173 − 0.984i)17-s + (1.39 + 0.245i)21-s + (1.39 − 0.245i)29-s + (0.483 + 1.32i)33-s + (0.766 + 0.642i)35-s + 1.41i·37-s + (0.909 − 1.08i)41-s + (0.939 − 0.342i)43-s + (0.499 + 0.866i)45-s + ⋯ |
L(s) = 1 | + (−0.909 + 1.08i)3-s + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)7-s + (−0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.483 − 1.32i)15-s + (0.173 − 0.984i)17-s + (1.39 + 0.245i)21-s + (1.39 − 0.245i)29-s + (0.483 + 1.32i)33-s + (0.766 + 0.642i)35-s + 1.41i·37-s + (0.909 − 1.08i)41-s + (0.939 − 0.342i)43-s + (0.499 + 0.866i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5527035481\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5527035481\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.909 - 1.08i)T + (-0.173 - 0.984i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (-1.39 + 0.245i)T + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 + (-0.909 + 1.08i)T + (-0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 59 | \( 1 + (1.39 + 0.245i)T + (0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (0.483 + 1.32i)T + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (0.939 + 0.342i)T^{2} \) |
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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
−9.907986420764233491246993007909, −9.065019587788052415263148853992, −8.040895052874293892488232812874, −7.14010723373297638118689353836, −6.40200757507579820510886989513, −5.44872590080901432395967857384, −4.47764294243128945348665231339, −3.84208467239671716717627394613, −3.05602487638446396071454604385, −0.60234090160508946529768394078,
1.18872135141460663158234252029, 2.47239845984761214581065418652, 3.88744766739949347227775719979, 4.80358021270631337859934608562, 5.96372369061554460994726869877, 6.36385363333305876303252482760, 7.34973757676725664543100045242, 7.945879951877690470074969176289, 8.871430546550454667649301731901, 9.685345587813347444910709860008