| L(s) = 1 | − 2.38·3-s + 0.0992·5-s − 0.236·7-s + 2.66·9-s + 2.12·11-s − 5.09·13-s − 0.236·15-s + 6.44·17-s + 0.582·19-s + 0.562·21-s − 4.99·25-s + 0.793·27-s + 3.65·29-s − 3.91·31-s − 5.04·33-s − 0.0234·35-s − 7.09·37-s + 12.1·39-s − 7.57·41-s + 10.7·43-s + 0.264·45-s + 10.8·47-s − 6.94·49-s − 15.3·51-s − 6.54·53-s + 0.210·55-s − 1.38·57-s + ⋯ |
| L(s) = 1 | − 1.37·3-s + 0.0444·5-s − 0.0893·7-s + 0.888·9-s + 0.639·11-s − 1.41·13-s − 0.0610·15-s + 1.56·17-s + 0.133·19-s + 0.122·21-s − 0.998·25-s + 0.152·27-s + 0.678·29-s − 0.702·31-s − 0.878·33-s − 0.00396·35-s − 1.16·37-s + 1.94·39-s − 1.18·41-s + 1.63·43-s + 0.0394·45-s + 1.58·47-s − 0.992·49-s − 2.14·51-s − 0.898·53-s + 0.0283·55-s − 0.183·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8937593116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8937593116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.38T + 3T^{2} \) |
| 5 | \( 1 - 0.0992T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 - 0.582T + 19T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 + 7.09T + 37T^{2} \) |
| 41 | \( 1 + 7.57T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 6.54T + 53T^{2} \) |
| 59 | \( 1 - 8.89T + 59T^{2} \) |
| 61 | \( 1 - 4.79T + 61T^{2} \) |
| 67 | \( 1 - 3.29T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 3.90T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 - 3.04T + 89T^{2} \) |
| 97 | \( 1 + 4.49T + 97T^{2} \) |
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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
−8.319689335090615863752147785850, −7.37357319915301328541596892151, −6.98247364130471146312202578489, −5.99434498923729594739325811588, −5.52624884564907690140386991777, −4.87026735447531687836162759471, −3.98580812884585133120473195113, −2.97574248764090913341213896426, −1.71656875808820771994633036876, −0.58068238686505894696412901157,
0.58068238686505894696412901157, 1.71656875808820771994633036876, 2.97574248764090913341213896426, 3.98580812884585133120473195113, 4.87026735447531687836162759471, 5.52624884564907690140386991777, 5.99434498923729594739325811588, 6.98247364130471146312202578489, 7.37357319915301328541596892151, 8.319689335090615863752147785850
