L(s) = 1 | − 2-s − 2.03·3-s + 4-s + 3.12·5-s + 2.03·6-s − 4.32·7-s − 8-s + 1.13·9-s − 3.12·10-s − 11-s − 2.03·12-s − 2.76·13-s + 4.32·14-s − 6.36·15-s + 16-s + 0.311·17-s − 1.13·18-s + 2.52·19-s + 3.12·20-s + 8.78·21-s + 22-s − 4.26·23-s + 2.03·24-s + 4.78·25-s + 2.76·26-s + 3.78·27-s − 4.32·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.17·3-s + 0.5·4-s + 1.39·5-s + 0.830·6-s − 1.63·7-s − 0.353·8-s + 0.379·9-s − 0.989·10-s − 0.301·11-s − 0.587·12-s − 0.765·13-s + 1.15·14-s − 1.64·15-s + 0.250·16-s + 0.0755·17-s − 0.268·18-s + 0.579·19-s + 0.699·20-s + 1.91·21-s + 0.213·22-s − 0.889·23-s + 0.415·24-s + 0.957·25-s + 0.541·26-s + 0.728·27-s − 0.816·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4641777802\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4641777802\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
good | 3 | \( 1 + 2.03T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 7 | \( 1 + 4.32T + 7T^{2} \) |
| 13 | \( 1 + 2.76T + 13T^{2} \) |
| 17 | \( 1 - 0.311T + 17T^{2} \) |
| 19 | \( 1 - 2.52T + 19T^{2} \) |
| 23 | \( 1 + 4.26T + 23T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 + 9.13T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 4.95T + 41T^{2} \) |
| 43 | \( 1 + 5.49T + 43T^{2} \) |
| 47 | \( 1 + 5.87T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 - 9.98T + 59T^{2} \) |
| 61 | \( 1 - 7.19T + 61T^{2} \) |
| 67 | \( 1 + 8.12T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.38T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 + 7.18T + 89T^{2} \) |
| 97 | \( 1 + 9.69T + 97T^{2} \) |
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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
−8.611942033754916281872145163864, −7.36109115640956864493467796255, −6.79632688921188472043388127128, −6.19135388446302090125970842284, −5.59496461253194744731089321364, −5.14200531877744665713992056437, −3.62871246508135529630474461118, −2.69808305853803020260734727678, −1.80788617679971419508678803504, −0.43280186104744185773414332008,
0.43280186104744185773414332008, 1.80788617679971419508678803504, 2.69808305853803020260734727678, 3.62871246508135529630474461118, 5.14200531877744665713992056437, 5.59496461253194744731089321364, 6.19135388446302090125970842284, 6.79632688921188472043388127128, 7.36109115640956864493467796255, 8.611942033754916281872145163864