L(s) = 1 | + 0.618·2-s − 1.61·4-s − 2.23·5-s + 7-s − 2.23·8-s − 1.38·10-s − 4.23·13-s + 0.618·14-s + 1.85·16-s − 2·17-s + 7.47·19-s + 3.61·20-s + 2·23-s − 2.61·26-s − 1.61·28-s − 5·29-s + 1.52·31-s + 5.61·32-s − 1.23·34-s − 2.23·35-s + 7.47·37-s + 4.61·38-s + 5.00·40-s + 6.47·41-s + 0.472·43-s + 1.23·46-s − 0.527·47-s + ⋯ |
L(s) = 1 | + 0.437·2-s − 0.809·4-s − 0.999·5-s + 0.377·7-s − 0.790·8-s − 0.437·10-s − 1.17·13-s + 0.165·14-s + 0.463·16-s − 0.485·17-s + 1.71·19-s + 0.809·20-s + 0.417·23-s − 0.513·26-s − 0.305·28-s − 0.928·29-s + 0.274·31-s + 0.993·32-s − 0.211·34-s − 0.377·35-s + 1.22·37-s + 0.749·38-s + 0.790·40-s + 1.01·41-s + 0.0720·43-s + 0.182·46-s − 0.0769·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 7.47T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 - 0.472T + 43T^{2} \) |
| 47 | \( 1 + 0.527T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 - 9.94T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 - 8.47T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.49998766188780965785890376124, −7.09504191888256671674590640983, −5.85069627749114426552617409701, −5.32656563552552108396634140195, −4.55171299638589287352044279562, −4.12486850470002047361570542622, −3.26868848174133350674425819847, −2.51662328984033359975041980738, −1.04740794424560128598230650101, 0,
1.04740794424560128598230650101, 2.51662328984033359975041980738, 3.26868848174133350674425819847, 4.12486850470002047361570542622, 4.55171299638589287352044279562, 5.32656563552552108396634140195, 5.85069627749114426552617409701, 7.09504191888256671674590640983, 7.49998766188780965785890376124