L(s) = 1 | − 1.56·2-s + 3-s + 0.438·4-s + 5-s − 1.56·6-s + 7-s + 2.43·8-s + 9-s − 1.56·10-s + 0.438·12-s + 5.56·13-s − 1.56·14-s + 15-s − 4.68·16-s + 4.12·17-s − 1.56·18-s + 6·19-s + 0.438·20-s + 21-s − 4·23-s + 2.43·24-s − 4·25-s − 8.68·26-s + 27-s + 0.438·28-s + 6.68·29-s − 1.56·30-s + ⋯ |
L(s) = 1 | − 1.10·2-s + 0.577·3-s + 0.219·4-s + 0.447·5-s − 0.637·6-s + 0.377·7-s + 0.862·8-s + 0.333·9-s − 0.493·10-s + 0.126·12-s + 1.54·13-s − 0.417·14-s + 0.258·15-s − 1.17·16-s + 0.999·17-s − 0.368·18-s + 1.37·19-s + 0.0980·20-s + 0.218·21-s − 0.834·23-s + 0.497·24-s − 0.800·25-s − 1.70·26-s + 0.192·27-s + 0.0828·28-s + 1.24·29-s − 0.285·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.600869753\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.600869753\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 - 5.56T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 - 3.43T + 47T^{2} \) |
| 53 | \( 1 - 7.80T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 8.80T + 67T^{2} \) |
| 71 | \( 1 - 3.12T + 71T^{2} \) |
| 73 | \( 1 - 7.12T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 8.80T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 1.31T + 97T^{2} \) |
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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
−8.808791178186706025377966303338, −8.193320392382910427986031826087, −7.83752791360058466545958936186, −6.82315556932055083436163382516, −5.92331233632029478084409522523, −4.98499229255871837576969402770, −3.96545678245895010800091965970, −3.04336360006953761694513366437, −1.70840803898572090112120367954, −1.03934814603548413725743981788,
1.03934814603548413725743981788, 1.70840803898572090112120367954, 3.04336360006953761694513366437, 3.96545678245895010800091965970, 4.98499229255871837576969402770, 5.92331233632029478084409522523, 6.82315556932055083436163382516, 7.83752791360058466545958936186, 8.193320392382910427986031826087, 8.808791178186706025377966303338