L(s) = 1 | − 2.80·2-s + 5.88·4-s − 5-s + 7-s − 10.9·8-s + 2.80·10-s + 2.07·11-s − 5.69·13-s − 2.80·14-s + 18.8·16-s + 3.22·17-s − 3.29·19-s − 5.88·20-s − 5.83·22-s + 3.22·23-s + 25-s + 15.9·26-s + 5.88·28-s + 9.77·29-s − 0.396·31-s − 31.1·32-s − 9.04·34-s − 35-s + 7.61·37-s + 9.26·38-s + 10.9·40-s − 9.99·41-s + ⋯ |
L(s) = 1 | − 1.98·2-s + 2.94·4-s − 0.447·5-s + 0.377·7-s − 3.85·8-s + 0.888·10-s + 0.626·11-s − 1.57·13-s − 0.750·14-s + 4.72·16-s + 0.781·17-s − 0.756·19-s − 1.31·20-s − 1.24·22-s + 0.671·23-s + 0.200·25-s + 3.13·26-s + 1.11·28-s + 1.81·29-s − 0.0711·31-s − 5.51·32-s − 1.55·34-s − 0.169·35-s + 1.25·37-s + 1.50·38-s + 1.72·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5722350504\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5722350504\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 - 3.22T + 23T^{2} \) |
| 29 | \( 1 - 9.77T + 29T^{2} \) |
| 31 | \( 1 + 0.396T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 + 9.99T + 41T^{2} \) |
| 43 | \( 1 - 2.60T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 5.61T + 59T^{2} \) |
| 61 | \( 1 + 1.22T + 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 - 3.69T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 6.39T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−9.986693872260891257775670236872, −9.193939240226480776357771541241, −8.399534768479077599675162055843, −7.75381116686615631872667836438, −7.01504509627684399848489245223, −6.27201680063656869747919254524, −4.87168406933367139638899198053, −3.18750182489063981503494188642, −2.13188469695963530738113665950, −0.792626379523688210897945799582,
0.792626379523688210897945799582, 2.13188469695963530738113665950, 3.18750182489063981503494188642, 4.87168406933367139638899198053, 6.27201680063656869747919254524, 7.01504509627684399848489245223, 7.75381116686615631872667836438, 8.399534768479077599675162055843, 9.193939240226480776357771541241, 9.986693872260891257775670236872