| L(s) = 1 | − 0.287·3-s − 0.115·5-s + 7-s − 2.91·9-s − 11-s − 13-s + 0.0331·15-s − 5.96·17-s + 2.56·19-s − 0.287·21-s + 1.63·23-s − 4.98·25-s + 1.69·27-s − 4.27·29-s + 9.90·31-s + 0.287·33-s − 0.115·35-s + 3.82·37-s + 0.287·39-s − 6.84·41-s + 2.08·43-s + 0.336·45-s − 10.6·47-s + 49-s + 1.71·51-s + 13.3·53-s + 0.115·55-s + ⋯ |
| L(s) = 1 | − 0.165·3-s − 0.0515·5-s + 0.377·7-s − 0.972·9-s − 0.301·11-s − 0.277·13-s + 0.00854·15-s − 1.44·17-s + 0.587·19-s − 0.0626·21-s + 0.340·23-s − 0.997·25-s + 0.327·27-s − 0.793·29-s + 1.77·31-s + 0.0500·33-s − 0.0194·35-s + 0.629·37-s + 0.0459·39-s − 1.06·41-s + 0.317·43-s + 0.0501·45-s − 1.55·47-s + 0.142·49-s + 0.239·51-s + 1.83·53-s + 0.0155·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.225740629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.225740629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.287T + 3T^{2} \) |
| 5 | \( 1 + 0.115T + 5T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 - 9.90T + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 6.75T + 67T^{2} \) |
| 71 | \( 1 + 6.49T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 2.81T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−7.79691982197706446658471107866, −7.23372633077182900042622144952, −6.32083975183587140331654591106, −5.83459131102295828766725505075, −4.95131408776866685346831099511, −4.48466770163022266044523523515, −3.43497087503946330687466578912, −2.64070180904266859657864631726, −1.87719523238672362857946798080, −0.53188743677340069151099207308,
0.53188743677340069151099207308, 1.87719523238672362857946798080, 2.64070180904266859657864631726, 3.43497087503946330687466578912, 4.48466770163022266044523523515, 4.95131408776866685346831099511, 5.83459131102295828766725505075, 6.32083975183587140331654591106, 7.23372633077182900042622144952, 7.79691982197706446658471107866
