L(s) = 1 | + (1 + i)3-s + (1.41 − 1.41i)5-s + 2.82i·7-s − i·9-s + (−3 + 3i)11-s + (4.24 + 4.24i)13-s + 2.82·15-s + (−3 − 3i)19-s + (−2.82 + 2.82i)21-s + 8.48i·23-s + 0.999i·25-s + (4 − 4i)27-s + (−1.41 − 1.41i)29-s + 5.65·31-s − 6·33-s + ⋯ |
L(s) = 1 | + (0.577 + 0.577i)3-s + (0.632 − 0.632i)5-s + 1.06i·7-s − 0.333i·9-s + (−0.904 + 0.904i)11-s + (1.17 + 1.17i)13-s + 0.730·15-s + (−0.688 − 0.688i)19-s + (−0.617 + 0.617i)21-s + 1.76i·23-s + 0.199i·25-s + (0.769 − 0.769i)27-s + (−0.262 − 0.262i)29-s + 1.01·31-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.093971517\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.093971517\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-1 - i)T + 3iT^{2} \) |
| 5 | \( 1 + (-1.41 + 1.41i)T - 5iT^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + (3 - 3i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.24 - 4.24i)T + 13iT^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (3 + 3i)T + 19iT^{2} \) |
| 23 | \( 1 - 8.48iT - 23T^{2} \) |
| 29 | \( 1 + (1.41 + 1.41i)T + 29iT^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (-4.24 + 4.24i)T - 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-1.41 + 1.41i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1 + i)T - 59iT^{2} \) |
| 61 | \( 1 + (4.24 + 4.24i)T + 61iT^{2} \) |
| 67 | \( 1 + (9 + 9i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 + (-3 - 3i)T + 83iT^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.654822164746029188403560033555, −9.321401860354949323932137199838, −8.766360360254689132419705584883, −7.83608134691249338907157688545, −6.56864706026007790513466908856, −5.74444130709378783949020129863, −4.84358600687926353850728758610, −3.92788843908594427768656399212, −2.67790057204657631290922216015, −1.67835338470405641873883802288,
0.937011587342487563751588173336, 2.43994079387302554807603276406, 3.16480341182307468886879174733, 4.39749495769089950646338537079, 5.75422565662984091614672075572, 6.39079065524980281887193049417, 7.38315732255965417090866813692, 8.236964474150488463668329479708, 8.536275703018802954253829879501, 10.16037624057318541419599902183