| L(s) = 1 | − 2.32·2-s − 1.14·3-s + 3.40·4-s + 2.67·6-s + 0.143·7-s − 3.26·8-s − 1.68·9-s − 2.81·11-s − 3.90·12-s − 1.70·13-s − 0.333·14-s + 0.778·16-s − 3.55·17-s + 3.90·18-s − 0.164·21-s + 6.54·22-s − 7.19·23-s + 3.74·24-s + 3.97·26-s + 5.37·27-s + 0.487·28-s + 7.57·29-s + 4.84·31-s + 4.71·32-s + 3.23·33-s + 8.27·34-s − 5.72·36-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 0.663·3-s + 1.70·4-s + 1.09·6-s + 0.0541·7-s − 1.15·8-s − 0.560·9-s − 0.848·11-s − 1.12·12-s − 0.473·13-s − 0.0890·14-s + 0.194·16-s − 0.863·17-s + 0.920·18-s − 0.0359·21-s + 1.39·22-s − 1.50·23-s + 0.765·24-s + 0.778·26-s + 1.03·27-s + 0.0921·28-s + 1.40·29-s + 0.870·31-s + 0.833·32-s + 0.562·33-s + 1.41·34-s − 0.953·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08923034432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08923034432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 + 1.14T + 3T^{2} \) |
| 7 | \( 1 - 0.143T + 7T^{2} \) |
| 11 | \( 1 + 2.81T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 23 | \( 1 + 7.19T + 23T^{2} \) |
| 29 | \( 1 - 7.57T + 29T^{2} \) |
| 31 | \( 1 - 4.84T + 31T^{2} \) |
| 37 | \( 1 + 9.49T + 37T^{2} \) |
| 41 | \( 1 + 0.187T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 + 5.36T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 6.00T + 67T^{2} \) |
| 71 | \( 1 - 0.540T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 7.59T + 83T^{2} \) |
| 89 | \( 1 - 1.44T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−8.050276679233652278649298457337, −7.09327433844816813763294314328, −6.60855554750227177973663019422, −5.92678393462454698017222607459, −5.07576408288522977025480907142, −4.41615576970521345117407810778, −3.03737588766173776314118477175, −2.36647823869587887944328411769, −1.46490967489741073786130747505, −0.19289220431034225860891724145,
0.19289220431034225860891724145, 1.46490967489741073786130747505, 2.36647823869587887944328411769, 3.03737588766173776314118477175, 4.41615576970521345117407810778, 5.07576408288522977025480907142, 5.92678393462454698017222607459, 6.60855554750227177973663019422, 7.09327433844816813763294314328, 8.050276679233652278649298457337
