L(s) = 1 | + (0.0395 − 0.829i)2-s + (0.308 + 0.0294i)4-s + (0.154 − 1.07i)8-s + (0.723 + 0.690i)9-s + (−0.0135 − 0.284i)11-s + (−0.583 − 0.112i)16-s + (0.601 − 0.573i)18-s − 0.236·22-s + (0.235 + 0.971i)23-s + (−0.888 + 0.458i)25-s + (0.698 − 1.53i)29-s + (0.140 − 0.577i)32-s + (0.202 + 0.234i)36-s + (−0.947 − 0.903i)37-s + (0.273 + 1.89i)43-s + (0.00419 − 0.0880i)44-s + ⋯ |
L(s) = 1 | + (0.0395 − 0.829i)2-s + (0.308 + 0.0294i)4-s + (0.154 − 1.07i)8-s + (0.723 + 0.690i)9-s + (−0.0135 − 0.284i)11-s + (−0.583 − 0.112i)16-s + (0.601 − 0.573i)18-s − 0.236·22-s + (0.235 + 0.971i)23-s + (−0.888 + 0.458i)25-s + (0.698 − 1.53i)29-s + (0.140 − 0.577i)32-s + (0.202 + 0.234i)36-s + (−0.947 − 0.903i)37-s + (0.273 + 1.89i)43-s + (0.00419 − 0.0880i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.285668533\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.285668533\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 + (-0.235 - 0.971i)T \) |
good | 2 | \( 1 + (-0.0395 + 0.829i)T + (-0.995 - 0.0950i)T^{2} \) |
| 3 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 5 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 11 | \( 1 + (0.0135 + 0.284i)T + (-0.995 + 0.0950i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 19 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 29 | \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \) |
| 31 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 37 | \( 1 + (0.947 + 0.903i)T + (0.0475 + 0.998i)T^{2} \) |
| 41 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.550 + 1.58i)T + (-0.786 + 0.618i)T^{2} \) |
| 59 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 61 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 67 | \( 1 + (0.738 - 0.380i)T + (0.580 - 0.814i)T^{2} \) |
| 71 | \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 79 | \( 1 + (0.550 - 1.58i)T + (-0.786 - 0.618i)T^{2} \) |
| 83 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 89 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 97 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
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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
−9.938001645305985970910312147216, −9.465946580612271701276824165309, −8.124318726520200113600791579840, −7.47012236112958185019533608741, −6.59280897792903391545380777516, −5.58777880067447771452915634262, −4.42082076361806063042748298138, −3.54989513395630151619421941132, −2.43835345939361919911694599421, −1.44441034242080142547721480980,
1.65863389572604380925206584982, 2.97783182769029753540712505267, 4.25994892481523761611678831496, 5.17444472513250723455186481333, 6.19161233661502624664497394315, 6.84716941914708108131723240812, 7.45329774665637324894705531357, 8.446479966335109446233501904671, 9.143344593131077133672280061931, 10.29291855147589213373584157531