| L(s) = 1 | + 3-s − 5-s + 3.33·7-s + 9-s − 1.83·11-s − 1.66·13-s − 15-s + 6.84·17-s − 6.80·19-s + 3.33·21-s − 4.13·23-s + 25-s + 27-s + 3.52·29-s + 5.55·31-s − 1.83·33-s − 3.33·35-s − 2.86·37-s − 1.66·39-s + 10.4·41-s + 8.71·43-s − 45-s + 4.81·47-s + 4.14·49-s + 6.84·51-s + 13.7·53-s + 1.83·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.26·7-s + 0.333·9-s − 0.552·11-s − 0.460·13-s − 0.258·15-s + 1.66·17-s − 1.56·19-s + 0.728·21-s − 0.861·23-s + 0.200·25-s + 0.192·27-s + 0.654·29-s + 0.998·31-s − 0.319·33-s − 0.564·35-s − 0.470·37-s − 0.266·39-s + 1.63·41-s + 1.32·43-s − 0.149·45-s + 0.702·47-s + 0.592·49-s + 0.959·51-s + 1.88·53-s + 0.247·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.508673326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.508673326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 - 3.33T + 7T^{2} \) |
| 11 | \( 1 + 1.83T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 + 4.13T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 - 5.55T + 31T^{2} \) |
| 37 | \( 1 + 2.86T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 8.71T + 43T^{2} \) |
| 47 | \( 1 - 4.81T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 + 2.29T + 59T^{2} \) |
| 61 | \( 1 - 6.17T + 61T^{2} \) |
| 71 | \( 1 + 7.19T + 71T^{2} \) |
| 73 | \( 1 + 5.29T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 9.17T + 83T^{2} \) |
| 89 | \( 1 + 7.13T + 89T^{2} \) |
| 97 | \( 1 - 1.89T + 97T^{2} \) |
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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
−8.429547736727951848788208404083, −7.64474550066335612075442941039, −7.45344296453099842650616616189, −6.17441120429397624703944140925, −5.40214966285181466848433992098, −4.49012173700042142396677143806, −4.01889257690425157284507114580, −2.81470225961021442085179187972, −2.10223663885880934064809269300, −0.906174268822109083688411882235,
0.906174268822109083688411882235, 2.10223663885880934064809269300, 2.81470225961021442085179187972, 4.01889257690425157284507114580, 4.49012173700042142396677143806, 5.40214966285181466848433992098, 6.17441120429397624703944140925, 7.45344296453099842650616616189, 7.64474550066335612075442941039, 8.429547736727951848788208404083
