| L(s) = 1 | + 0.979·2-s − 1.04·4-s − 1.37·5-s + 3.49·7-s − 2.97·8-s − 1.34·10-s − 2.07·11-s + 4.50·13-s + 3.42·14-s − 0.837·16-s + 1.29·17-s + 1.52·19-s + 1.42·20-s − 2.03·22-s + 3.75·23-s − 3.12·25-s + 4.41·26-s − 3.63·28-s − 0.108·29-s + 2.35·31-s + 5.13·32-s + 1.26·34-s − 4.79·35-s + 10.5·37-s + 1.49·38-s + 4.08·40-s + 9.87·41-s + ⋯ |
| L(s) = 1 | + 0.692·2-s − 0.520·4-s − 0.612·5-s + 1.32·7-s − 1.05·8-s − 0.424·10-s − 0.626·11-s + 1.25·13-s + 0.915·14-s − 0.209·16-s + 0.314·17-s + 0.350·19-s + 0.318·20-s − 0.433·22-s + 0.782·23-s − 0.624·25-s + 0.866·26-s − 0.687·28-s − 0.0202·29-s + 0.423·31-s + 0.907·32-s + 0.217·34-s − 0.809·35-s + 1.72·37-s + 0.242·38-s + 0.645·40-s + 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.995590676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995590676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 0.979T + 2T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 4.50T + 13T^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 19 | \( 1 - 1.52T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 + 0.108T + 29T^{2} \) |
| 31 | \( 1 - 2.35T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 9.87T + 41T^{2} \) |
| 43 | \( 1 - 2.45T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 + 2.28T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 + 2.08T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 6.08T + 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−9.735388771001088165322408268412, −8.843547975579932902373798342942, −8.086241303930166954433870774439, −7.58531036393446209127754000853, −6.13352767102694421494516104342, −5.40649093756567537522133964171, −4.52473554927437949552866832571, −3.89369495745188145199539740771, −2.74304353399344584116455816928, −1.03216374256784898566455495636,
1.03216374256784898566455495636, 2.74304353399344584116455816928, 3.89369495745188145199539740771, 4.52473554927437949552866832571, 5.40649093756567537522133964171, 6.13352767102694421494516104342, 7.58531036393446209127754000853, 8.086241303930166954433870774439, 8.843547975579932902373798342942, 9.735388771001088165322408268412
