L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 1.61·7-s − 8-s + 9-s + 0.618·11-s − 12-s + 1.85·13-s + 1.61·14-s + 16-s − 5.23·17-s − 18-s − 0.854·19-s + 1.61·21-s − 0.618·22-s + 1.85·23-s + 24-s − 1.85·26-s − 27-s − 1.61·28-s − 7.23·29-s + 6.47·31-s − 32-s − 0.618·33-s + 5.23·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.611·7-s − 0.353·8-s + 0.333·9-s + 0.186·11-s − 0.288·12-s + 0.514·13-s + 0.432·14-s + 0.250·16-s − 1.26·17-s − 0.235·18-s − 0.195·19-s + 0.353·21-s − 0.131·22-s + 0.386·23-s + 0.204·24-s − 0.363·26-s − 0.192·27-s − 0.305·28-s − 1.34·29-s + 1.16·31-s − 0.176·32-s − 0.107·33-s + 0.897·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 0.618T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 - 1.85T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 - 0.618T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 + 1.90T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 + 6.90T + 89T^{2} \) |
| 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
−9.951473967070916097461052877444, −9.086836040638722481835154795515, −8.398030197634529302795165248625, −7.15073673323062541880957356011, −6.56743490124000774670703675385, −5.68736614296276151981811890139, −4.43505676390634117498281156287, −3.19552408320630230696342379817, −1.70611611768860655788354343983, 0,
1.70611611768860655788354343983, 3.19552408320630230696342379817, 4.43505676390634117498281156287, 5.68736614296276151981811890139, 6.56743490124000774670703675385, 7.15073673323062541880957356011, 8.398030197634529302795165248625, 9.086836040638722481835154795515, 9.951473967070916097461052877444