| L(s) = 1 | + 0.618·3-s − 1.61·7-s − 2.61·9-s − 1.23·11-s + 1.23·13-s − 7.23·17-s + 3.23·19-s − 1.00·21-s − 5.61·23-s − 3.47·27-s − 3.85·29-s + 2.76·31-s − 0.763·33-s − 6·37-s + 0.763·39-s − 2.14·41-s + 4.32·43-s − 9.09·47-s − 4.38·49-s − 4.47·51-s + 4.94·53-s + 2.00·57-s + 14.1·59-s − 1.90·61-s + 4.23·63-s − 10.4·67-s − 3.47·69-s + ⋯ |
| L(s) = 1 | + 0.356·3-s − 0.611·7-s − 0.872·9-s − 0.372·11-s + 0.342·13-s − 1.75·17-s + 0.742·19-s − 0.218·21-s − 1.17·23-s − 0.668·27-s − 0.715·29-s + 0.496·31-s − 0.132·33-s − 0.986·37-s + 0.122·39-s − 0.335·41-s + 0.659·43-s − 1.32·47-s − 0.625·49-s − 0.626·51-s + 0.679·53-s + 0.264·57-s + 1.84·59-s − 0.244·61-s + 0.533·63-s − 1.27·67-s − 0.417·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 5.61T + 23T^{2} \) |
| 29 | \( 1 + 3.85T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 2.14T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 + 9.09T + 47T^{2} \) |
| 53 | \( 1 - 4.94T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 1.90T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.478546220884378982311107920409, −8.718890156299550977539205108514, −8.062714314894758965580973894265, −6.98099193116547492878832181185, −6.16541566686016816677313328158, −5.28113515024793653408877187194, −4.06053105773645338264422611956, −3.09326583955386921041093125982, −2.08227788744085250702839123532, 0,
2.08227788744085250702839123532, 3.09326583955386921041093125982, 4.06053105773645338264422611956, 5.28113515024793653408877187194, 6.16541566686016816677313328158, 6.98099193116547492878832181185, 8.062714314894758965580973894265, 8.718890156299550977539205108514, 9.478546220884378982311107920409
