| L(s) = 1 | − 2-s − 1.90·3-s + 4-s − 3.72·5-s + 1.90·6-s + 7-s − 8-s + 0.623·9-s + 3.72·10-s + 2.91·11-s − 1.90·12-s + 0.858·13-s − 14-s + 7.09·15-s + 16-s + 2.98·17-s − 0.623·18-s − 1.77·19-s − 3.72·20-s − 1.90·21-s − 2.91·22-s + 7.03·23-s + 1.90·24-s + 8.89·25-s − 0.858·26-s + 4.52·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.09·3-s + 0.5·4-s − 1.66·5-s + 0.777·6-s + 0.377·7-s − 0.353·8-s + 0.207·9-s + 1.17·10-s + 0.878·11-s − 0.549·12-s + 0.238·13-s − 0.267·14-s + 1.83·15-s + 0.250·16-s + 0.722·17-s − 0.147·18-s − 0.407·19-s − 0.833·20-s − 0.415·21-s − 0.621·22-s + 1.46·23-s + 0.388·24-s + 1.77·25-s − 0.168·26-s + 0.870·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6772615214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6772615214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.90T + 3T^{2} \) |
| 5 | \( 1 + 3.72T + 5T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 - 0.858T + 13T^{2} \) |
| 17 | \( 1 - 2.98T + 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 23 | \( 1 - 7.03T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 0.802T + 31T^{2} \) |
| 37 | \( 1 - 0.206T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 1.00T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 + 5.01T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 8.51T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 0.847T + 73T^{2} \) |
| 79 | \( 1 - 2.38T + 79T^{2} \) |
| 83 | \( 1 + 1.46T + 83T^{2} \) |
| 89 | \( 1 - 8.13T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.011512782986212187759427531283, −7.48945787216117534116435178797, −6.61240753989330448568909088942, −6.27490062533388993457822847553, −5.07606592170859752078034710296, −4.59574394461885736713944237665, −3.64152711009353948815770814738, −2.88658200443108940496288354350, −1.26040203489935925823022716461, −0.59610364601031808992820841491,
0.59610364601031808992820841491, 1.26040203489935925823022716461, 2.88658200443108940496288354350, 3.64152711009353948815770814738, 4.59574394461885736713944237665, 5.07606592170859752078034710296, 6.27490062533388993457822847553, 6.61240753989330448568909088942, 7.48945787216117534116435178797, 8.011512782986212187759427531283
