| L(s) = 1 | + i·2-s − 4-s + 2.34·5-s + (1.07 + 2.41i)7-s − i·8-s + 2.34i·10-s + 5.67i·11-s − 1.71i·13-s + (−2.41 + 1.07i)14-s + 16-s + 1.76·17-s + 1.13i·19-s − 2.34·20-s − 5.67·22-s − 3.67i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.05·5-s + (0.407 + 0.913i)7-s − 0.353i·8-s + 0.742i·10-s + 1.71i·11-s − 0.477i·13-s + (−0.645 + 0.288i)14-s + 0.250·16-s + 0.429·17-s + 0.261i·19-s − 0.525·20-s − 1.21·22-s − 0.766i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.856948066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.856948066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.07 - 2.41i)T \) |
| good | 5 | \( 1 - 2.34T + 5T^{2} \) |
| 11 | \( 1 - 5.67iT - 11T^{2} \) |
| 13 | \( 1 + 1.71iT - 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 - 1.13iT - 19T^{2} \) |
| 23 | \( 1 + 3.67iT - 23T^{2} \) |
| 29 | \( 1 + 4.15iT - 29T^{2} \) |
| 31 | \( 1 - 8.37iT - 31T^{2} \) |
| 37 | \( 1 + 9.19T + 37T^{2} \) |
| 41 | \( 1 - 7.99T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 - 8.99iT - 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 4.52iT - 71T^{2} \) |
| 73 | \( 1 + 5.34iT - 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 + 4.59iT - 97T^{2} \) |
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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
−9.997393913591396561304360265663, −9.186481128162343937222456108824, −8.484368275297951523788194646429, −7.51531822309151793776597408253, −6.72666431570743034822573767704, −5.74251514993785165273328401174, −5.22641679562659197562609627718, −4.26932248717629533912599724840, −2.64437977367793050065619021444, −1.67318351653426810046563007841,
0.857797693006781451123261729050, 1.97656993578732109411043202170, 3.24176733616498206097477587719, 4.09887237019612837037939333769, 5.34607586131042199687986658344, 5.94257854542717655622363174415, 7.07805131687448432074096357387, 8.089894738198778589911148587298, 8.935759705641816727522455785936, 9.660578052876508185548332975780
