L(s) = 1 | + 2-s − 1.24·3-s + 4-s − 5-s − 1.24·6-s + 0.198·7-s + 8-s − 1.44·9-s − 10-s + 1.10·11-s − 1.24·12-s + 0.198·14-s + 1.24·15-s + 16-s + 6.49·17-s − 1.44·18-s − 0.493·19-s − 20-s − 0.246·21-s + 1.10·22-s − 6.76·23-s − 1.24·24-s + 25-s + 5.54·27-s + 0.198·28-s − 0.356·29-s + 1.24·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.719·3-s + 0.5·4-s − 0.447·5-s − 0.509·6-s + 0.0748·7-s + 0.353·8-s − 0.481·9-s − 0.316·10-s + 0.334·11-s − 0.359·12-s + 0.0529·14-s + 0.321·15-s + 0.250·16-s + 1.57·17-s − 0.340·18-s − 0.113·19-s − 0.223·20-s − 0.0538·21-s + 0.236·22-s − 1.41·23-s − 0.254·24-s + 0.200·25-s + 1.06·27-s + 0.0374·28-s − 0.0662·29-s + 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.887218949\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.887218949\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 3 | \( 1 + 1.24T + 3T^{2} \) |
| 7 | \( 1 - 0.198T + 7T^{2} \) |
| 11 | \( 1 - 1.10T + 11T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 + 0.493T + 19T^{2} \) |
| 23 | \( 1 + 6.76T + 23T^{2} \) |
| 29 | \( 1 + 0.356T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 - 2.21T + 37T^{2} \) |
| 41 | \( 1 - 8.96T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 - 9.92T + 53T^{2} \) |
| 59 | \( 1 - 4.61T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 + 3.87T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 0.591T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−9.504652064421652428771534421350, −8.274000178564659722318610597240, −7.76667075382333317358651753383, −6.70020079171363120772752405399, −5.94794372798999430692609971969, −5.37280822643577229181416280433, −4.37455610283489034456909878654, −3.56560822348763525776280257450, −2.48707585069931253013461803269, −0.902450970279651450566763189984,
0.902450970279651450566763189984, 2.48707585069931253013461803269, 3.56560822348763525776280257450, 4.37455610283489034456909878654, 5.37280822643577229181416280433, 5.94794372798999430692609971969, 6.70020079171363120772752405399, 7.76667075382333317358651753383, 8.274000178564659722318610597240, 9.504652064421652428771534421350