L(s) = 1 | + (0.311 − 2.21i)5-s − i·7-s − 5.05·11-s + 3.37i·13-s + 7.18i·17-s + 8.23·19-s − 6.23i·23-s + (−4.80 − 1.37i)25-s − 2·29-s + 4.62·31-s + (−2.21 − 0.311i)35-s + 4.85i·37-s + 3.37·41-s − 1.24i·43-s − 49-s + ⋯ |
L(s) = 1 | + (0.139 − 0.990i)5-s − 0.377i·7-s − 1.52·11-s + 0.936i·13-s + 1.74i·17-s + 1.88·19-s − 1.30i·23-s + (−0.961 − 0.275i)25-s − 0.371·29-s + 0.830·31-s + (−0.374 − 0.0525i)35-s + 0.798i·37-s + 0.527·41-s − 0.189i·43-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.718661119\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.718661119\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.311 + 2.21i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 5.05T + 11T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 - 7.18iT - 17T^{2} \) |
| 19 | \( 1 - 8.23T + 19T^{2} \) |
| 23 | \( 1 + 6.23iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 - 4.85iT - 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 1.24iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 4.62iT - 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 0.488T + 61T^{2} \) |
| 67 | \( 1 - 3.61iT - 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 16.2iT - 73T^{2} \) |
| 79 | \( 1 - 1.24T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 6.99T + 89T^{2} \) |
| 97 | \( 1 - 8.23iT - 97T^{2} \) |
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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
−8.330850297151758965179303893726, −7.63276575698985869009084608034, −6.82355448992549180555176295875, −5.90281607053642112444514704521, −5.29130063545028492510322365181, −4.55828081334318631966481376787, −3.87519645325334476631229374457, −2.77014186231784471523167540722, −1.78890466710255442303103820699, −0.78130618201775824429196768471,
0.64510706570011428912176030109, 2.19262103795353320096434977034, 3.03581364024175795597727159146, 3.26976486560009151568396484309, 4.79894967523562607294768478955, 5.47800850552834606036612528249, 5.83564438766407093002697255097, 7.06232189079156751803920490086, 7.62233937099821466473617299638, 7.83915119288428216306098890546