L(s) = 1 | + 0.618·3-s − 2.32·7-s − 2.61·9-s + 5.56·11-s + 5.76·13-s + 0.711·17-s − 7.15·19-s − 1.43·21-s + 0.906·23-s − 3.47·27-s − 5.23·29-s − 4.47·31-s + 3.43·33-s − 8.24·37-s + 3.56·39-s + 1.05·41-s + 2.76·43-s − 0.560·47-s − 1.57·49-s + 0.439·51-s + 8.47·53-s − 4.42·57-s + 9.94·59-s + 14.0·61-s + 6.09·63-s − 2.82·67-s + 0.560·69-s + ⋯ |
L(s) = 1 | + 0.356·3-s − 0.880·7-s − 0.872·9-s + 1.67·11-s + 1.60·13-s + 0.172·17-s − 1.64·19-s − 0.314·21-s + 0.188·23-s − 0.668·27-s − 0.972·29-s − 0.803·31-s + 0.598·33-s − 1.35·37-s + 0.571·39-s + 0.165·41-s + 0.420·43-s − 0.0816·47-s − 0.224·49-s + 0.0616·51-s + 1.16·53-s − 0.585·57-s + 1.29·59-s + 1.80·61-s + 0.768·63-s − 0.344·67-s + 0.0674·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 - 5.76T + 13T^{2} \) |
| 17 | \( 1 - 0.711T + 17T^{2} \) |
| 19 | \( 1 + 7.15T + 19T^{2} \) |
| 23 | \( 1 - 0.906T + 23T^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 1.05T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 + 0.560T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 9.94T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 + 7.91T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 5.76T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 + 8.50T + 97T^{2} \) |
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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
−7.06780300373545178013738646847, −6.68791423615248123615249481889, −5.93718602511439846881731605813, −5.59015791879992188041981626276, −4.14237666358822470530569047177, −3.80904450176876793520947410267, −3.19694734818075751984747298436, −2.15119640099294174958029969636, −1.28329529387591255234836991924, 0,
1.28329529387591255234836991924, 2.15119640099294174958029969636, 3.19694734818075751984747298436, 3.80904450176876793520947410267, 4.14237666358822470530569047177, 5.59015791879992188041981626276, 5.93718602511439846881731605813, 6.68791423615248123615249481889, 7.06780300373545178013738646847