| L(s) = 1 | − 0.434·3-s + 3.15·7-s − 2.81·9-s + 5.02·11-s + 4.08·13-s − 7.55·17-s + 2.65·19-s − 1.37·21-s + 7.18·23-s + 2.52·27-s − 29-s + 4.72·31-s − 2.18·33-s − 5.98·37-s − 1.77·39-s − 9.58·41-s + 5.74·43-s + 0.565·47-s + 2.96·49-s + 3.28·51-s + 10.2·53-s − 1.15·57-s − 1.33·59-s + 2.15·61-s − 8.87·63-s + 9.69·67-s − 3.12·69-s + ⋯ |
| L(s) = 1 | − 0.251·3-s + 1.19·7-s − 0.936·9-s + 1.51·11-s + 1.13·13-s − 1.83·17-s + 0.610·19-s − 0.299·21-s + 1.49·23-s + 0.486·27-s − 0.185·29-s + 0.848·31-s − 0.380·33-s − 0.984·37-s − 0.284·39-s − 1.49·41-s + 0.876·43-s + 0.0824·47-s + 0.423·49-s + 0.460·51-s + 1.41·53-s − 0.153·57-s − 0.174·59-s + 0.275·61-s − 1.11·63-s + 1.18·67-s − 0.376·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.091399666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091399666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 0.434T + 3T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 5.02T + 11T^{2} \) |
| 13 | \( 1 - 4.08T + 13T^{2} \) |
| 17 | \( 1 + 7.55T + 17T^{2} \) |
| 19 | \( 1 - 2.65T + 19T^{2} \) |
| 23 | \( 1 - 7.18T + 23T^{2} \) |
| 31 | \( 1 - 4.72T + 31T^{2} \) |
| 37 | \( 1 + 5.98T + 37T^{2} \) |
| 41 | \( 1 + 9.58T + 41T^{2} \) |
| 43 | \( 1 - 5.74T + 43T^{2} \) |
| 47 | \( 1 - 0.565T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 1.33T + 59T^{2} \) |
| 61 | \( 1 - 2.15T + 61T^{2} \) |
| 67 | \( 1 - 9.69T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 2.78T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 0.504T + 83T^{2} \) |
| 89 | \( 1 - 8.79T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.817925011099870147400312866264, −8.279435263630472146034255089934, −7.06240905640744003753165485760, −6.56135148311863692770860971957, −5.68827560673029538840527164347, −4.86999282045119503296223936083, −4.12914397172662364243904936132, −3.15724790650940818459654847820, −1.91607185870830579605802774698, −0.962337808388013687616438397297,
0.962337808388013687616438397297, 1.91607185870830579605802774698, 3.15724790650940818459654847820, 4.12914397172662364243904936132, 4.86999282045119503296223936083, 5.68827560673029538840527164347, 6.56135148311863692770860971957, 7.06240905640744003753165485760, 8.279435263630472146034255089934, 8.817925011099870147400312866264
