L(s) = 1 | + 2.31·2-s + 3.37·4-s − 4.06·5-s − 2.19·7-s + 3.18·8-s − 9.41·10-s − 1.39·11-s − 3.98·13-s − 5.09·14-s + 0.628·16-s − 3.51·17-s + 19-s − 13.6·20-s − 3.23·22-s − 6.58·23-s + 11.4·25-s − 9.23·26-s − 7.41·28-s − 2.87·29-s + 8.99·31-s − 4.90·32-s − 8.14·34-s + 8.92·35-s + 7.82·37-s + 2.31·38-s − 12.9·40-s + 11.1·41-s + ⋯ |
L(s) = 1 | + 1.63·2-s + 1.68·4-s − 1.81·5-s − 0.830·7-s + 1.12·8-s − 2.97·10-s − 0.420·11-s − 1.10·13-s − 1.36·14-s + 0.157·16-s − 0.852·17-s + 0.229·19-s − 3.06·20-s − 0.689·22-s − 1.37·23-s + 2.29·25-s − 1.81·26-s − 1.40·28-s − 0.533·29-s + 1.61·31-s − 0.867·32-s − 1.39·34-s + 1.50·35-s + 1.28·37-s + 0.376·38-s − 2.04·40-s + 1.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.955415400\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.955415400\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 7 | \( 1 + 2.19T + 7T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 13 | \( 1 + 3.98T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 23 | \( 1 + 6.58T + 23T^{2} \) |
| 29 | \( 1 + 2.87T + 29T^{2} \) |
| 31 | \( 1 - 8.99T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 53 | \( 1 + 2.08T + 53T^{2} \) |
| 59 | \( 1 - 9.16T + 59T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 - 6.65T + 67T^{2} \) |
| 71 | \( 1 + 7.43T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 8.98T + 83T^{2} \) |
| 89 | \( 1 + 4.08T + 89T^{2} \) |
| 97 | \( 1 - 1.02T + 97T^{2} \) |
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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
−7.68925133244948776417750685872, −6.98136545770534744404691336905, −6.39549335098675690840204359318, −5.62817245906086117470861925347, −4.75476569605757056219224932401, −4.14860477883498377410594622132, −3.88709157798472165127906378898, −2.77184408585502185404854158029, −2.52658329208597201061007325804, −0.49926122514731731830036920038,
0.49926122514731731830036920038, 2.52658329208597201061007325804, 2.77184408585502185404854158029, 3.88709157798472165127906378898, 4.14860477883498377410594622132, 4.75476569605757056219224932401, 5.62817245906086117470861925347, 6.39549335098675690840204359318, 6.98136545770534744404691336905, 7.68925133244948776417750685872