L(s) = 1 | − 2-s − 0.942·3-s + 4-s + 5-s + 0.942·6-s + 2.99·7-s − 8-s − 2.11·9-s − 10-s + 5.03·11-s − 0.942·12-s + 13-s − 2.99·14-s − 0.942·15-s + 16-s − 3.38·17-s + 2.11·18-s − 7.01·19-s + 20-s − 2.81·21-s − 5.03·22-s − 3.52·23-s + 0.942·24-s + 25-s − 26-s + 4.81·27-s + 2.99·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.543·3-s + 0.5·4-s + 0.447·5-s + 0.384·6-s + 1.13·7-s − 0.353·8-s − 0.704·9-s − 0.316·10-s + 1.51·11-s − 0.271·12-s + 0.277·13-s − 0.799·14-s − 0.243·15-s + 0.250·16-s − 0.821·17-s + 0.497·18-s − 1.60·19-s + 0.223·20-s − 0.614·21-s − 1.07·22-s − 0.734·23-s + 0.192·24-s + 0.200·25-s − 0.196·26-s + 0.926·27-s + 0.565·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.370993744\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.370993744\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 + 0.942T + 3T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 17 | \( 1 + 3.38T + 17T^{2} \) |
| 19 | \( 1 + 7.01T + 19T^{2} \) |
| 23 | \( 1 + 3.52T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 0.208T + 41T^{2} \) |
| 43 | \( 1 - 1.78T + 43T^{2} \) |
| 47 | \( 1 + 1.05T + 47T^{2} \) |
| 53 | \( 1 - 3.32T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 0.791T + 73T^{2} \) |
| 79 | \( 1 + 4.83T + 79T^{2} \) |
| 83 | \( 1 - 4.96T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 - 4.87T + 97T^{2} \) |
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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
−8.469133069170981650841855276163, −8.013699387913579956032418163486, −6.69990732027180034871355081672, −6.42631728364245259923240708861, −5.70139475056982269353949557342, −4.64427787148251518992827822680, −4.03861947440229035448762304136, −2.58509164321605323395886283201, −1.79879027939932378918961809443, −0.789284814792921276207211398430,
0.789284814792921276207211398430, 1.79879027939932378918961809443, 2.58509164321605323395886283201, 4.03861947440229035448762304136, 4.64427787148251518992827822680, 5.70139475056982269353949557342, 6.42631728364245259923240708861, 6.69990732027180034871355081672, 8.013699387913579956032418163486, 8.469133069170981650841855276163