L(s) = 1 | + (1.20 − 1.88i)5-s + 1.28i·7-s + 4.14·11-s + 6.52i·13-s + 5.98i·17-s − 7.17·19-s + 7.53i·23-s + (−2.08 − 4.54i)25-s − 5.19·29-s + 5.17·31-s + (2.41 + 1.54i)35-s − 5.24i·37-s + 0.680·41-s − 1.28i·43-s + 5.31i·47-s + ⋯ |
L(s) = 1 | + (0.539 − 0.841i)5-s + 0.484i·7-s + 1.24·11-s + 1.80i·13-s + 1.45i·17-s − 1.64·19-s + 1.57i·23-s + (−0.417 − 0.908i)25-s − 0.964·29-s + 0.930·31-s + (0.407 + 0.261i)35-s − 0.861i·37-s + 0.106·41-s − 0.195i·43-s + 0.774i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.761622486\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.761622486\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.20 + 1.88i)T \) |
good | 7 | \( 1 - 1.28iT - 7T^{2} \) |
| 11 | \( 1 - 4.14T + 11T^{2} \) |
| 13 | \( 1 - 6.52iT - 13T^{2} \) |
| 17 | \( 1 - 5.98iT - 17T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 23 | \( 1 - 7.53iT - 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 - 5.17T + 31T^{2} \) |
| 37 | \( 1 + 5.24iT - 37T^{2} \) |
| 41 | \( 1 - 0.680T + 41T^{2} \) |
| 43 | \( 1 + 1.28iT - 43T^{2} \) |
| 47 | \( 1 - 5.31iT - 47T^{2} \) |
| 53 | \( 1 + 2.21iT - 53T^{2} \) |
| 59 | \( 1 - 7.60T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 - 15.6iT - 67T^{2} \) |
| 71 | \( 1 - 5.50T + 71T^{2} \) |
| 73 | \( 1 + 7.80iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 9.75iT - 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 14.3iT - 97T^{2} \) |
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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
−9.220102897874785902268084517939, −8.968019439700660822057936588120, −8.188510171178844143797657442442, −6.92518887192118667273539512217, −6.22430948109238017627248490291, −5.59187360632661997391824649673, −4.26228315442408976776066086420, −3.97279057206456251125364689685, −2.08462541596955301521802529456, −1.52450434883807959672339501111,
0.69088774826727342543938868381, 2.27501439770638048899578206733, 3.14467889491210168575978120433, 4.15866815103282713181269267239, 5.18613124192674343689488815715, 6.26492590797414836336687331473, 6.70140013968376806847301218703, 7.60021915846258482957916544557, 8.486700827313580217636806634410, 9.341554447042751008508003270940