L(s) = 1 | − 1.39·5-s + 4.25·7-s + 3.66·11-s − 2.28·13-s + 5.79·17-s + 4.93·19-s + 1.43·23-s − 3.06·25-s + 3.08·29-s − 7.83·31-s − 5.92·35-s + 1.91·37-s + 1.76·41-s + 0.276·43-s + 9.79·47-s + 11.1·49-s + 10.1·53-s − 5.09·55-s + 13.3·59-s − 5.62·61-s + 3.18·65-s − 9.11·67-s − 13.3·71-s + 3.42·73-s + 15.5·77-s + 5.57·79-s − 7.04·83-s + ⋯ |
L(s) = 1 | − 0.622·5-s + 1.60·7-s + 1.10·11-s − 0.634·13-s + 1.40·17-s + 1.13·19-s + 0.300·23-s − 0.612·25-s + 0.572·29-s − 1.40·31-s − 1.00·35-s + 0.314·37-s + 0.275·41-s + 0.0421·43-s + 1.42·47-s + 1.58·49-s + 1.39·53-s − 0.686·55-s + 1.73·59-s − 0.719·61-s + 0.394·65-s − 1.11·67-s − 1.58·71-s + 0.401·73-s + 1.77·77-s + 0.627·79-s − 0.773·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.573453468\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.573453468\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 + 2.28T + 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 + 7.83T + 31T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 41 | \( 1 - 1.76T + 41T^{2} \) |
| 43 | \( 1 - 0.276T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 5.62T + 61T^{2} \) |
| 67 | \( 1 + 9.11T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 3.42T + 73T^{2} \) |
| 79 | \( 1 - 5.57T + 79T^{2} \) |
| 83 | \( 1 + 7.04T + 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.86721322359554443508434711640, −7.54151979060927856002258237214, −6.93714966578250704465546329906, −5.62230995296548501648976592084, −5.34547720660093261112291884168, −4.31353688878121553234707107717, −3.85184024635726378942180995246, −2.80736936092075934964945395079, −1.65490934222848407330032856447, −0.930422722283315848559123778629,
0.930422722283315848559123778629, 1.65490934222848407330032856447, 2.80736936092075934964945395079, 3.85184024635726378942180995246, 4.31353688878121553234707107717, 5.34547720660093261112291884168, 5.62230995296548501648976592084, 6.93714966578250704465546329906, 7.54151979060927856002258237214, 7.86721322359554443508434711640