L(s) = 1 | − 2.31·2-s + 0.311·3-s + 3.35·4-s + 1.29·5-s − 0.721·6-s − 1.13·7-s − 3.13·8-s − 2.90·9-s − 2.99·10-s + 2.78·11-s + 1.04·12-s + 3.93·13-s + 2.63·14-s + 0.403·15-s + 0.546·16-s − 3.61·17-s + 6.71·18-s + 4.23·19-s + 4.34·20-s − 0.354·21-s − 6.45·22-s − 0.383·23-s − 0.977·24-s − 3.32·25-s − 9.10·26-s − 1.84·27-s − 3.81·28-s + ⋯ |
L(s) = 1 | − 1.63·2-s + 0.179·3-s + 1.67·4-s + 0.579·5-s − 0.294·6-s − 0.429·7-s − 1.10·8-s − 0.967·9-s − 0.948·10-s + 0.840·11-s + 0.301·12-s + 1.09·13-s + 0.703·14-s + 0.104·15-s + 0.136·16-s − 0.876·17-s + 1.58·18-s + 0.971·19-s + 0.972·20-s − 0.0773·21-s − 1.37·22-s − 0.0798·23-s − 0.199·24-s − 0.664·25-s − 1.78·26-s − 0.354·27-s − 0.721·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 - 0.311T + 3T^{2} \) |
| 5 | \( 1 - 1.29T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 + 0.383T + 23T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 3.45T + 37T^{2} \) |
| 41 | \( 1 - 2.80T + 41T^{2} \) |
| 43 | \( 1 - 4.55T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 0.380T + 53T^{2} \) |
| 59 | \( 1 - 6.87T + 59T^{2} \) |
| 61 | \( 1 - 2.49T + 61T^{2} \) |
| 67 | \( 1 + 9.83T + 67T^{2} \) |
| 71 | \( 1 - 3.25T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−8.273206034948867536290013664806, −7.58687870361909594189251101024, −6.70493484775521317900727480701, −6.16592759905646871992048284573, −5.44082752666945920171886560029, −4.01629162008560446206130714235, −3.11546216693094901110308539535, −2.09881655930972765332948333634, −1.29363917265238692410684199450, 0,
1.29363917265238692410684199450, 2.09881655930972765332948333634, 3.11546216693094901110308539535, 4.01629162008560446206130714235, 5.44082752666945920171886560029, 6.16592759905646871992048284573, 6.70493484775521317900727480701, 7.58687870361909594189251101024, 8.273206034948867536290013664806