| L(s) = 1 | − 2.04·2-s + 2.19·4-s + 3.35·5-s − 2.24·7-s − 0.405·8-s − 6.87·10-s + 4.93·11-s + 4.60·14-s − 3.56·16-s − 0.911·17-s + 3.80·19-s + 7.37·20-s − 10.1·22-s − 2.02·23-s + 6.26·25-s − 4.93·28-s + 3.93·29-s + 8.82·31-s + 8.11·32-s + 1.86·34-s − 7.54·35-s − 8.80·37-s − 7.78·38-s − 1.36·40-s + 6.93·41-s − 2.28·43-s + 10.8·44-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.09·4-s + 1.50·5-s − 0.849·7-s − 0.143·8-s − 2.17·10-s + 1.48·11-s + 1.23·14-s − 0.891·16-s − 0.221·17-s + 0.872·19-s + 1.64·20-s − 2.15·22-s − 0.422·23-s + 1.25·25-s − 0.933·28-s + 0.731·29-s + 1.58·31-s + 1.43·32-s + 0.320·34-s − 1.27·35-s − 1.44·37-s − 1.26·38-s − 0.215·40-s + 1.08·41-s − 0.348·43-s + 1.63·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.094261412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.094261412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 5 | \( 1 - 3.35T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 - 4.93T + 11T^{2} \) |
| 17 | \( 1 + 0.911T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 + 8.80T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 + 2.28T + 43T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 + 0.542T + 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 + 7.41T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 2.30T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−9.403772098341645772727020107405, −9.033095982790242372753993766949, −8.141113226307116693295325589601, −6.91830238092506950036906206160, −6.53656168646630609396376229148, −5.71816619123980213229469750834, −4.42940888803132794871732855147, −3.03801430934366866862812944157, −1.90476316612678443500821202247, −0.981247455175690461555243617867,
0.981247455175690461555243617867, 1.90476316612678443500821202247, 3.03801430934366866862812944157, 4.42940888803132794871732855147, 5.71816619123980213229469750834, 6.53656168646630609396376229148, 6.91830238092506950036906206160, 8.141113226307116693295325589601, 9.033095982790242372753993766949, 9.403772098341645772727020107405
