L(s) = 1 | − 3.28·3-s + (1.54 + 1.62i)5-s + 2.73i·7-s + 7.80·9-s − 2.28·11-s − 4.92i·13-s + (−5.06 − 5.32i)15-s − 5.97·17-s + 0.377·19-s − 8.97i·21-s + (−0.582 − 4.76i)23-s + (−0.249 + 4.99i)25-s − 15.7·27-s + 3.70·29-s + 2.89i·31-s + ⋯ |
L(s) = 1 | − 1.89·3-s + (0.689 + 0.724i)5-s + 1.03i·7-s + 2.60·9-s − 0.689·11-s − 1.36i·13-s + (−1.30 − 1.37i)15-s − 1.44·17-s + 0.0866·19-s − 1.95i·21-s + (−0.121 − 0.992i)23-s + (−0.0498 + 0.998i)25-s − 3.03·27-s + 0.687·29-s + 0.519i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4859673530\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4859673530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.54 - 1.62i)T \) |
| 23 | \( 1 + (0.582 + 4.76i)T \) |
good | 3 | \( 1 + 3.28T + 3T^{2} \) |
| 7 | \( 1 - 2.73iT - 7T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + 4.92iT - 13T^{2} \) |
| 17 | \( 1 + 5.97T + 17T^{2} \) |
| 19 | \( 1 - 0.377T + 19T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 2.89iT - 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 - 0.451T + 41T^{2} \) |
| 43 | \( 1 + 0.359iT - 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 + 10.7iT - 59T^{2} \) |
| 61 | \( 1 + 6.73iT - 61T^{2} \) |
| 67 | \( 1 - 7.06iT - 67T^{2} \) |
| 71 | \( 1 + 7.34iT - 71T^{2} \) |
| 73 | \( 1 - 1.24iT - 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 15.1iT - 83T^{2} \) |
| 89 | \( 1 - 12.1iT - 89T^{2} \) |
| 97 | \( 1 - 4.34T + 97T^{2} \) |
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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
−9.305411751877733337673245077099, −8.282378857754218013033732834551, −7.13728366662655126589780225281, −6.50653095216341741961658816839, −5.81334263534504028537376351378, −5.32299116819886091210889339318, −4.55027030114032080626033481042, −2.97060791211220351248227769432, −1.94240413634637535208103659533, −0.27669064038406783530399436821,
0.991727455453452374264848053441, 2.00051803706002772340216313106, 4.16212406752966801102389770542, 4.54362472846113090012579541687, 5.39822378065730008730175722449, 6.13745958142022369641901886277, 6.87471456485965882309645297770, 7.44667312166225291483006635443, 8.809985578916694201813996941713, 9.609115981668690490951006657572