| L(s) = 1 | + 0.296·3-s − 3.46·7-s − 2.91·9-s + 3.11·11-s + 4.60·13-s + 5.49·17-s − 4.48·19-s − 1.02·21-s − 23-s − 1.75·27-s + 9.19·29-s + 5.89·31-s + 0.924·33-s − 6.95·37-s + 1.36·39-s − 9.03·41-s + 5.55·43-s − 5.48·47-s + 4.97·49-s + 1.63·51-s − 2.74·53-s − 1.33·57-s − 9.33·59-s − 1.40·61-s + 10.0·63-s − 3.49·67-s − 0.296·69-s + ⋯ |
| L(s) = 1 | + 0.171·3-s − 1.30·7-s − 0.970·9-s + 0.938·11-s + 1.27·13-s + 1.33·17-s − 1.02·19-s − 0.224·21-s − 0.208·23-s − 0.337·27-s + 1.70·29-s + 1.05·31-s + 0.160·33-s − 1.14·37-s + 0.218·39-s − 1.41·41-s + 0.846·43-s − 0.800·47-s + 0.710·49-s + 0.228·51-s − 0.376·53-s − 0.176·57-s − 1.21·59-s − 0.179·61-s + 1.26·63-s − 0.427·67-s − 0.0357·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.740109876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740109876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.296T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 - 5.49T + 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 29 | \( 1 - 9.19T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 41 | \( 1 + 9.03T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 + 5.48T + 47T^{2} \) |
| 53 | \( 1 + 2.74T + 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 + 3.49T + 67T^{2} \) |
| 71 | \( 1 + 4.28T + 71T^{2} \) |
| 73 | \( 1 - 4.92T + 73T^{2} \) |
| 79 | \( 1 + 2.12T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 4.37T + 97T^{2} \) |
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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
−7.904126174824514748511508079374, −6.71951824857133705222912017440, −6.36426162253076280991584429787, −5.95817589798468117334164736899, −4.98185192683007112808691928309, −4.00367677841926572641470908329, −3.30955460721640162476139312797, −2.95113986089432960771244194822, −1.67764915935313273386006203788, −0.63618330069817038771388726737,
0.63618330069817038771388726737, 1.67764915935313273386006203788, 2.95113986089432960771244194822, 3.30955460721640162476139312797, 4.00367677841926572641470908329, 4.98185192683007112808691928309, 5.95817589798468117334164736899, 6.36426162253076280991584429787, 6.71951824857133705222912017440, 7.904126174824514748511508079374
