L(s) = 1 | + 5-s + 2.38·7-s − 5.33·11-s − 4.53·13-s − 1.81·17-s + 7.00·19-s + 23-s + 25-s + 0.118·29-s − 0.884·31-s + 2.38·35-s + 7.51·37-s + 1.45·41-s − 10.4·47-s − 1.32·49-s + 9.42·53-s − 5.33·55-s − 7.79·59-s − 2.80·61-s − 4.53·65-s − 3.11·67-s + 13.5·71-s + 12.4·73-s − 12.7·77-s + 6.80·79-s + 13.5·83-s − 1.81·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.900·7-s − 1.60·11-s − 1.25·13-s − 0.440·17-s + 1.60·19-s + 0.208·23-s + 0.200·25-s + 0.0219·29-s − 0.158·31-s + 0.402·35-s + 1.23·37-s + 0.226·41-s − 1.52·47-s − 0.189·49-s + 1.29·53-s − 0.719·55-s − 1.01·59-s − 0.359·61-s − 0.562·65-s − 0.380·67-s + 1.61·71-s + 1.45·73-s − 1.44·77-s + 0.765·79-s + 1.48·83-s − 0.196·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.020589064\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.020589064\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 + 1.81T + 17T^{2} \) |
| 19 | \( 1 - 7.00T + 19T^{2} \) |
| 29 | \( 1 - 0.118T + 29T^{2} \) |
| 31 | \( 1 + 0.884T + 31T^{2} \) |
| 37 | \( 1 - 7.51T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 + 2.80T + 61T^{2} \) |
| 67 | \( 1 + 3.11T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 2.89T + 89T^{2} \) |
| 97 | \( 1 + 1.97T + 97T^{2} \) |
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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
−7.75439918450227469210137170157, −7.35566462744005159835792049977, −6.42986413431045574438234186874, −5.47480578126739390116645691687, −5.05060444451028939113916043167, −4.60844251710145976019409731613, −3.31359731956540054699556897469, −2.55790502968397206883568069156, −1.92278472834604443669717788447, −0.68406165400172018254651081345,
0.68406165400172018254651081345, 1.92278472834604443669717788447, 2.55790502968397206883568069156, 3.31359731956540054699556897469, 4.60844251710145976019409731613, 5.05060444451028939113916043167, 5.47480578126739390116645691687, 6.42986413431045574438234186874, 7.35566462744005159835792049977, 7.75439918450227469210137170157