L(s) = 1 | + 2-s − 2.29·3-s + 4-s + 1.21·5-s − 2.29·6-s − 7-s + 8-s + 2.26·9-s + 1.21·10-s + 4.50·11-s − 2.29·12-s − 3.62·13-s − 14-s − 2.79·15-s + 16-s + 1.79·17-s + 2.26·18-s + 7.76·19-s + 1.21·20-s + 2.29·21-s + 4.50·22-s + 1.60·23-s − 2.29·24-s − 3.51·25-s − 3.62·26-s + 1.68·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.32·3-s + 0.5·4-s + 0.544·5-s − 0.936·6-s − 0.377·7-s + 0.353·8-s + 0.755·9-s + 0.385·10-s + 1.35·11-s − 0.662·12-s − 1.00·13-s − 0.267·14-s − 0.721·15-s + 0.250·16-s + 0.435·17-s + 0.533·18-s + 1.78·19-s + 0.272·20-s + 0.500·21-s + 0.959·22-s + 0.334·23-s − 0.468·24-s − 0.703·25-s − 0.711·26-s + 0.324·27-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
\(L(1)\) |
\(\approx\) |
\(2.248748938\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.248748938\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 - 1.21T + 5T^{2} \) |
| 11 | \( 1 - 4.50T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 - 1.79T + 17T^{2} \) |
| 19 | \( 1 - 7.76T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 + 0.942T + 31T^{2} \) |
| 37 | \( 1 + 2.03T + 37T^{2} \) |
| 41 | \( 1 - 3.33T + 41T^{2} \) |
| 43 | \( 1 + 7.76T + 43T^{2} \) |
| 47 | \( 1 + 2.12T + 47T^{2} \) |
| 53 | \( 1 + 4.29T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 0.965T + 61T^{2} \) |
| 67 | \( 1 + 8.00T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 9.74T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 5.12T + 97T^{2} \) |
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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
−7.73606151400242822995220723206, −7.01167472629353497321087203285, −6.51459369790731826089331635532, −5.83242851595632689047002053961, −5.28768464659709345834994718639, −4.72369329428771666874901980598, −3.73126476638023577158888562089, −2.95419728919539980584471399768, −1.74075770046637284073929598432, −0.78468350033145355920368432947,
0.78468350033145355920368432947, 1.74075770046637284073929598432, 2.95419728919539980584471399768, 3.73126476638023577158888562089, 4.72369329428771666874901980598, 5.28768464659709345834994718639, 5.83242851595632689047002053961, 6.51459369790731826089331635532, 7.01167472629353497321087203285, 7.73606151400242822995220723206