L(s) = 1 | + 1.79i·3-s + 4.79i·7-s − 0.208·9-s − 11-s − i·13-s + 3.79i·17-s + 2.58·19-s − 8.58·21-s + 0.791i·23-s + 5.00i·27-s − 2.20·29-s + 0.582·31-s − 1.79i·33-s − 6.58i·37-s + 1.79·39-s + ⋯ |
L(s) = 1 | + 1.03i·3-s + 1.81i·7-s − 0.0695·9-s − 0.301·11-s − 0.277i·13-s + 0.919i·17-s + 0.592·19-s − 1.87·21-s + 0.164i·23-s + 0.962i·27-s − 0.410·29-s + 0.104·31-s − 0.311i·33-s − 1.08i·37-s + 0.286·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.390213349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.390213349\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 1.79iT - 3T^{2} \) |
| 7 | \( 1 - 4.79iT - 7T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 3.79iT - 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 0.791iT - 23T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 - 0.582T + 31T^{2} \) |
| 37 | \( 1 + 6.58iT - 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 2.37iT - 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 + 3.20iT - 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 - 10.7iT - 97T^{2} \) |
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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
−10.08141520383487296141256445795, −9.402893901059134096184535663764, −8.740041136030264167055978697142, −7.998960736920526174337529527975, −6.75360959831682662220650797476, −5.53162913909998488141051766085, −5.32207774160311750296445549845, −4.04929585101012358080694340617, −3.09275382058044285948534933225, −1.95458200588052628896219001414,
0.62501305905652533546536089243, 1.66072238751476767802702823504, 3.12619379043241550266004382526, 4.21864141573697743138806802541, 5.13668151145891497198652472129, 6.55553196354825656720438605706, 7.02510777925570319407286241155, 7.64300291594859008223193863154, 8.390910479282867117532091184312, 9.770589560513343747675033790546