| L(s) = 1 | − i·2-s − 4-s − 2.82i·5-s − 2.64i·7-s + i·8-s − 2.82·10-s + (2.64 − 2i)11-s + 5.65·13-s − 2.64·14-s + 16-s + 7.48·17-s − 2.82·19-s + 2.82i·20-s + (−2 − 2.64i)22-s + 5.29·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.26i·5-s − 0.999i·7-s + 0.353i·8-s − 0.894·10-s + (0.797 − 0.603i)11-s + 1.56·13-s − 0.707·14-s + 0.250·16-s + 1.81·17-s − 0.648·19-s + 0.632i·20-s + (−0.426 − 0.564i)22-s + 1.10·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.832926277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.832926277\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| 11 | \( 1 + (-2.64 + 2i)T \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 7.48T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 7.48iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 + 5.29iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 5.29T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 5.29iT - 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−9.032369459870050857593460052595, −8.747892388083589135282392607069, −7.929312509724290111387634191537, −6.78345992329501320046903534239, −5.73601059097033462919210832087, −4.90613315395936126564530817470, −3.86349426662569053080318838580, −3.39106934616950427853007332756, −1.37960647309059414548772509229, −0.935510447391173674593521508826,
1.59516352875141401837492155138, 3.10032691704054380442857380130, 3.73108791247670040645742355713, 5.13709078996049378063600340431, 5.95796968344108919816140296018, 6.62209245684033086612889629888, 7.26703048435793255783151423566, 8.318992746171354890626889945077, 8.923417130791307774024452355613, 9.816046822686424449582795208530
