L(s) = 1 | + 2-s + 2.24·3-s + 4-s + 1.69·5-s + 2.24·6-s − 7-s + 8-s + 2.04·9-s + 1.69·10-s + 0.445·11-s + 2.24·12-s − 14-s + 3.80·15-s + 16-s − 2.15·17-s + 2.04·18-s + 6.35·19-s + 1.69·20-s − 2.24·21-s + 0.445·22-s − 0.911·23-s + 2.24·24-s − 2.13·25-s − 2.13·27-s − 28-s + 3.58·29-s + 3.80·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.756·5-s + 0.917·6-s − 0.377·7-s + 0.353·8-s + 0.682·9-s + 0.535·10-s + 0.134·11-s + 0.648·12-s − 0.267·14-s + 0.981·15-s + 0.250·16-s − 0.523·17-s + 0.482·18-s + 1.45·19-s + 0.378·20-s − 0.490·21-s + 0.0948·22-s − 0.190·23-s + 0.458·24-s − 0.427·25-s − 0.411·27-s − 0.188·28-s + 0.665·29-s + 0.694·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.991135146\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.991135146\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 5 | \( 1 - 1.69T + 5T^{2} \) |
| 11 | \( 1 - 0.445T + 11T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 - 6.35T + 19T^{2} \) |
| 23 | \( 1 + 0.911T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 - 8.89T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 2.41T + 41T^{2} \) |
| 43 | \( 1 - 4.63T + 43T^{2} \) |
| 47 | \( 1 + 9.75T + 47T^{2} \) |
| 53 | \( 1 + 8.74T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 1.37T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 + 5.76T + 71T^{2} \) |
| 73 | \( 1 + 9.93T + 73T^{2} \) |
| 79 | \( 1 + 6.30T + 79T^{2} \) |
| 83 | \( 1 + 2.91T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 3.30T + 97T^{2} \) |
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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
−9.111037609855064053661027352030, −8.081853903830939703635402939416, −7.58677532408271765610362148939, −6.48511654029689579116965473402, −5.95982217573503243218932671641, −4.88145457634050711964668802649, −4.02012316849193116834220697501, −2.99465980874160581162293047811, −2.58060668054311888099501511624, −1.40850576997079563377082833895,
1.40850576997079563377082833895, 2.58060668054311888099501511624, 2.99465980874160581162293047811, 4.02012316849193116834220697501, 4.88145457634050711964668802649, 5.95982217573503243218932671641, 6.48511654029689579116965473402, 7.58677532408271765610362148939, 8.081853903830939703635402939416, 9.111037609855064053661027352030