L(s) = 1 | − 1.89i·2-s + i·3-s − 1.58·4-s + 1.86i·5-s + 1.89·6-s − 0.793i·8-s − 9-s + 3.52·10-s + (−1.71 + 2.84i)11-s − 1.58i·12-s − 0.900·13-s − 1.86·15-s − 4.66·16-s + 4.81·17-s + 1.89i·18-s + 0.291·19-s + ⋯ |
L(s) = 1 | − 1.33i·2-s + 0.577i·3-s − 0.790·4-s + 0.834i·5-s + 0.772·6-s − 0.280i·8-s − 0.333·9-s + 1.11·10-s + (−0.516 + 0.856i)11-s − 0.456i·12-s − 0.249·13-s − 0.481·15-s − 1.16·16-s + 1.16·17-s + 0.446i·18-s + 0.0667·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.130740110\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.130740110\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (1.71 - 2.84i)T \) |
good | 2 | \( 1 + 1.89iT - 2T^{2} \) |
| 5 | \( 1 - 1.86iT - 5T^{2} \) |
| 13 | \( 1 + 0.900T + 13T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 - 0.291T + 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 - 3.00iT - 29T^{2} \) |
| 31 | \( 1 - 2.50iT - 31T^{2} \) |
| 37 | \( 1 + 7.22T + 37T^{2} \) |
| 41 | \( 1 - 3.97T + 41T^{2} \) |
| 43 | \( 1 - 8.50iT - 43T^{2} \) |
| 47 | \( 1 - 11.5iT - 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 8.44iT - 59T^{2} \) |
| 61 | \( 1 + 2.88T + 61T^{2} \) |
| 67 | \( 1 - 1.24T + 67T^{2} \) |
| 71 | \( 1 + 2.01T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.26iT - 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 12.5iT - 89T^{2} \) |
| 97 | \( 1 - 7.48iT - 97T^{2} \) |
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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
−9.777387619223812208141252952812, −9.209180860049917772560477492067, −7.955313598624102178825854481480, −7.19255331512240677290071069237, −6.23225669293813920760529989104, −5.07894112608924716138655279165, −4.23663463838384901752472100178, −3.21820197452064119660634326492, −2.70678803755578457649343162074, −1.49536587250706833283612362565,
0.42763087539934366764814780608, 1.98418240437476937218989428021, 3.35683950352580033070766837492, 4.70546789358482515776590896258, 5.52087055812081925838806007665, 5.94993988216078962940678428018, 6.96490748401617676551146572988, 7.69961179128517041272744738615, 8.294427419160092277923313177168, 8.811792023413360619418534438835