L(s) = 1 | + (−0.173 + 0.300i)2-s + (−0.766 + 1.32i)3-s + (0.439 + 0.761i)4-s + (−0.266 − 0.460i)6-s − 0.652·8-s + (−0.673 − 1.16i)9-s − 1.34·12-s + (0.173 − 0.984i)13-s + (−0.326 + 0.565i)16-s + 0.467·18-s + (−0.5 + 0.866i)23-s + (0.499 − 0.866i)24-s + 25-s + (0.266 + 0.223i)26-s + 0.532·27-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.300i)2-s + (−0.766 + 1.32i)3-s + (0.439 + 0.761i)4-s + (−0.266 − 0.460i)6-s − 0.652·8-s + (−0.673 − 1.16i)9-s − 1.34·12-s + (0.173 − 0.984i)13-s + (−0.326 + 0.565i)16-s + 0.467·18-s + (−0.5 + 0.866i)23-s + (0.499 − 0.866i)24-s + 25-s + (0.266 + 0.223i)26-s + 0.532·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6048630564\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6048630564\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - 1.53T + T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - 0.347T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + 1.87T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−12.09912944720434286943098971458, −11.20495328884785150887558841119, −10.53861824374153608681044066522, −9.577181552910066877078428732206, −8.561198416329453223390443375627, −7.52707319949893869316405981145, −6.26841501003280740617300131626, −5.33486246122980898888786396052, −4.11741859963816061257159406817, −3.03023510607917327183621616453,
1.23281169933623035559214643207, 2.50025538278488658699854800608, 4.73494891595770197915854247051, 6.14418389111510558898225052194, 6.49320073020378711582648325897, 7.55425339329387746438202985215, 8.812453217818125826819962838554, 10.00041205199057679380821971156, 10.96825794848189479008452176368, 11.70830870281403651213063811156