| L(s) = 1 | + 2-s + 4-s − 1.48·5-s + 0.344·7-s + 8-s − 1.48·10-s + 0.905·11-s − 2.24·13-s + 0.344·14-s + 16-s − 2.58·17-s − 0.688·19-s − 1.48·20-s + 0.905·22-s − 1.46·23-s − 2.78·25-s − 2.24·26-s + 0.344·28-s + 4.24·29-s + 1.92·31-s + 32-s − 2.58·34-s − 0.512·35-s − 1.84·37-s − 0.688·38-s − 1.48·40-s − 1.21·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.665·5-s + 0.130·7-s + 0.353·8-s − 0.470·10-s + 0.273·11-s − 0.623·13-s + 0.0919·14-s + 0.250·16-s − 0.627·17-s − 0.157·19-s − 0.332·20-s + 0.193·22-s − 0.304·23-s − 0.557·25-s − 0.441·26-s + 0.0650·28-s + 0.787·29-s + 0.346·31-s + 0.176·32-s − 0.443·34-s − 0.0865·35-s − 0.303·37-s − 0.111·38-s − 0.235·40-s − 0.190·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 269 | \( 1 + T \) |
| good | 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 - 0.344T + 7T^{2} \) |
| 11 | \( 1 - 0.905T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 0.688T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 1.92T + 31T^{2} \) |
| 37 | \( 1 + 1.84T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 + 6.59T + 43T^{2} \) |
| 47 | \( 1 - 6.42T + 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 1.75T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 + 5.29T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 1.61T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.86757087508741780496800157791, −7.08410000109443907158557408088, −6.48784288022064439393501864435, −5.67621298850110808062172856317, −4.74441659112986400700392678630, −4.29307029482091240396776632351, −3.42285749406430796454312685294, −2.57732236074449970259088925085, −1.55447494543000927108292122236, 0,
1.55447494543000927108292122236, 2.57732236074449970259088925085, 3.42285749406430796454312685294, 4.29307029482091240396776632351, 4.74441659112986400700392678630, 5.67621298850110808062172856317, 6.48784288022064439393501864435, 7.08410000109443907158557408088, 7.86757087508741780496800157791
